The sentence means something like "As we both know, he is indeed going to Leipzig tomorrow morning".

The speaker uses "ja" to insist that what s/he is saying is correct, at the same time assuming that the listener shares the same opinion.  An English equivalent might be to use the "do" form of the verb or add "indeed", along with adding something like “I bet you’d agree with me that…” or “as we both know…”

This sentence corresponds to something like: "All the same, he is going to Leipzig (despite the fact that there are a few issues with the trip)".

"allerdings" adds a tone of reservation to what's just been said. It corresponds to something like the English ‘even so’, ‘of course’, ‘admittedly’, ‘to be sure’ or ‘all the same’.

The feeling expressed is something like "Wow, the car's going fast!  I expected the car to go fast, but it's going much faster than thought it would!"

"aber" in an expression expresses surprise at how far something goes, in contradiction to what was expected

The meaning is something like "Wow, the car is indeed going fast! (That's a big surprise to me -- I had expected it to go slowly)".

"ja" expresses surprise that something is the opposite of what was expected.

"doch" could have been used with the same effect

The sentence is saying something like "However he is in fact going to Leipzig tomorrow morning (in spite of the fact that you thought he wasn't)"

Within a clause, "aber" is used to express a contradiction.  "But", "however" and "though" are close equivalents in English.

"doch" could also have been used to create a similar mood

The feeling of this sentence with the modal particle is something like:  "So, in conclusion, he'll be travelling to Leipzig tomorrow morning."

With "also" the speaker is confirming that this is a logical conclusion from what was previously discussed.

Here the meaning is something like "You'd better take the  car, or else!"

"ja" intensifies a command, containing an implied warning or threat, especially if emphasized

Other alternative modal particles that would have had the same effect would be "nur" or "bloß"

The feel corresponds to something like "please" that softens the message, something like: "Could you please take the car."

"eben mal" could have been used to further soften the message ("Fahr eben mal mit dem Auto!")

"doch mal" could have been used to make the message sound more casual ("Fahr doch mal mit dem Auto!")

Assuming that the question is spoken in a neutral tone, the meaning is something like a politely spoken "Will you be travelling to Berlin?"

In a neutral context, "denn" tones down a question and make it less blunt. 

Here the meaning would be equivalent to something like "Tell me, are going to be heading off to Berlin?" (conveyed via a friendly, casual tone)

"eigentlich" tones down a yes/no question and makes it sound more casual

Here the meaning would be something like: "Don't tell me, you're going to Berlin!?" (We both know that that means....Why the hell are you going?!)

"etwa" implies that this something is very undesirable and the answer ought to be "no"!

Here the meaning of the response to the first person saying "I'm very tired today" has the feel of something like: "Well, why do you always go to bed so late?!", spoken in an irritated tone)

"auch" used in a w-question like this, makes it into a  rhetorical question (one expressed to make a point and not because the speaker expects an answer)

Da kann man halt nichts machen.

What does "halt" add to this statement?

What other modal particle(s) could have been used to the same effect?

The statement is saying something like "There is really nothing that can be done about that (shrug)"

In statements, eben/halt emphasize an inescapable conclusion, the fact that there is no real alternative.

"eben" has the exact same meaning as "halt" and they can replace each other in all contexts.

Da kommt ja Peter!

What does the modal particle "ja" add to this exclamation?

The meaning of this exclamation is something like: "Wow!  Peter's coming!! (What a surprise!)"

In exclamations, "ja" expresses surprise.

Dann machen Sie die Übung eben noch einmal!

What does the modal particle "eben" add to this command?

The meaning of this command is something like: "Well then do the exercise again! (There's no escaping it - you have no other choice!)"

In commands, "eben" emphasizes that there is no real alternative

The meaning of this statement is something like: "It might be rather surprising to some, but in fact I really like that about him."

In statements, eigentlich implies that something is actually the case, despite appearances.  Here, yes, I do like that characteristic...It wouldn't be to everyone's taste, but I do like it!

Das hättest du nicht tun sollen -- Ich habe es auch nicht getan!

What does the modal particle "auch" add to this statement?

What other modal particle would often be used with it in this context?

The meaning of this statement is something like: "You shouldn’t have done that.— And the fact is, I did not do it!"

In statements, auch stresses the reasons why something is or is not the case, approximated here with "the fact is". As here, "auch" can be used to correct a false impression.

"Ja" .... – Ich habe es ja auch nicht getan!

Das war ja früher alles ganz anders.

What does the modal particle "ja" add to this statement?

The feel of the sentence with the modal particle is something like: "Of course, things used to be quite different."

In statements, ja is used to insist that what the speaker is saying is correct, presupposing that the listener agrees.

Den Wagen kann ich mir doch morgen abholen? (the "doch" is not stressed)

What does the modal particle "doch" add to this question?

The meaning of this question is something like: "I can pick up the car tomorrow, can't I?"

Unstressed "doch" indicates uncertainty, and is sort of the equivalent of adding a tag question.

Die Suppe ist aber heiß!

What does the modal particle "aber" add to this exclamation?

In this case, the translation could be something like: "My word, this soup is hot! (I didn't expect it to be nearly this hot!)"

In statements and exclamations, "aber" can express surprise at the degree of something.

It can also be simply used for emphasis, something like "The soup is really hot!", without the element of surprise necessarily coming into play.

Du hättest das eigentlich nicht machen sollen.

What does "eigentlich" add to this statement?

The feel of this sentence with the modal particle is something like: "You shouldn't really have done that (it might have looked like a good idea on the surface, but really, it wasn't)"

In statements, eigentlich implies that something is actually the case, despite appearances.

Es regnet!  Nun, dann machen wir das Picknick eben drinnen (or you could also say " Nun, dann machen wir eben das Picknick drinnen")

What does the modal particle "eben" add to this statement?

The meaning of this statement is something like: "It's raining!  Well, we'll just have the picnic inside (as an obvious solution)"

In statements, "eben" emphasizes an inescapable conclusion.

It could be is saying something like "Would you please go there." or it could be saying something like "Go there right now! (Come on, what are you waiting for?)"

In commands, the force of "doch" varies greatly depending on the context and tone.  It can make the  command more advisory or encouraging, as in the first case, or it can express impatience or urgency.

In the case where the sentence is advisory or encouraging, "doch" would often be combined with  "mal" or "ruhig"  (i.e. "Gehen Sie doch mal hin" or "Gehen Sie doch ruhig hin").

Gerhard sieht sich heute schlecht aus – Er ist auch lange krank gewesen.

What does the modal particle "auch" add to this statement?

What other modal particle would often be used with it in this context?

The meaning of this statement is something like: "Gerhard is not looking well today – Well, the fact is, he’s been ill for a very long time….."

In statements, auch stresses the reasons why something is or is not the case, approximated here with "the fact is". "Auch" can be used to correct a false impression.

"Ja" .... – Er ist ja auch lange krank gewesen.

Hast du auch die Rechnung bezahlt?

If "auch" is being used as a modal particle here (rather than to mean English "also"), what does it add to this sentence?

The feel of this question with the modal particle is something like "You did pay the bill, didn't you? (I'm assuming that you did but am just checking!)"

"Auch" in a yes/no question asks for confirmation of something the speaker thinks is very likely to be the case and is just trying to make sure of. 

Hast du die Katze etwa schon wieder gefüttert?

What does the modal particle "etwa" add to this question?

The meaning of this question is something like: "Don't tell me you've already fed the cat again?!"

In yes/no questions, "etwa" implies that something is very undesireable, something like "Don't tell me that...."?!

Hast du ihm auch gesagt, dass wir nicht kömmen können?

What does the modal particle "auch" add to this question?

The feel of the sentence with the modal particle is something like: "You have told him we can't come, haven't you? (I'm assuming that you have, but I just thought I'd better check)."

In yes/no questions, "auch" asks for confirmation of something which the speaker thinks should be taken for granted. The English equivalent is very often a tag question.

Hast du was dagegen? Aber nein! 

What does the modal particle "aber" add to this exclamation?

The meaning of this statement is something like: "Have you any objections?  No, of course not!!!"

When used initially in exclamations, "aber" stresses the speaker’s opinion.

Ist das Wetter denn nicht herrlich?

What does "denn" add to this question?

In this case, meaning is something like "Wow, isn't the weather gorgious?!  (It's turned out to be much lovelier than I expected!)"

"denn" in a yes/no question can be used to express surprise

Könnten Sie mir mal das Buch dort geben?

What does "mal" do in this question?

The meaning of this question is something like:  "Could  you please give me that book?"

Whereever it's used, "mal" moderates the tone of a sentence, making it sound less blunt, and implying that the speaker just wants to bother the listener "for a sec".

Leg das nur wieder hin!

What role does "nur" play in this command?

Which other modal particle(s) could have been used to the same effect?

The command is saying something like "You’d better put that down or you’ll be sorry!"

"nur" intensifies the basic meaning of a command.  Depending on the basic meaning of the command, it can sound like more of a threat (as here), or more reassuring.

In a "threatening" or "warning" command like this one, "nur" could be replaced by "ja" or "bloß"

Mein Freund kam aber nicht.
 

The meaning of this statement is something like: "However my friend did not come."

Aber is used within a clause to express a contradiction.  This use of aber is close to "doch" and “but”, “however” or “though” are close equivalents in English. 

Monika sollte doch heute abend kommen. (with "doch" stressed)

What does the modal particle "doch" add to this question?

The meaning of this statement is something like: "Yes (in spite of what you think) Monika was supposed to come tonight!"

In statements, stressed "doch" indicates disagreement with what has been said.

Morgen soll es doch regnen, oder?

What does the modal particle "doch" add to this question?

The meaning of this question is something like: "It's supposed to rain tomorrow, isn't it?"

Unstressed "doch" indicates uncertainty, which is also expressed through the use of the "oder?"

The meaning of this command is something like: "Well then, don't smoke so much! (I mean, really! I'm irritated that you smoke so much despite the fact that you obviously shouldn't!)"

In commands, "doch" can be used in different ways, depending on the tone used, but in this case it adds a note of impatience

Sagen Sie das bloß nicht zu laut!

What does the modal particle "bloß" add to this exclamation?

What other modal particle might have been used.

The sentence is saying something like "Don’t you dare say that aloud!"

As a modal particle, bloß is often used in a warning. 

"Nur" could have been used. bloß is used in many of the same situations as nur (with exceptions) but it is a bit more emphatic and it is used only in less formal contexts.

Sei auch schon brav!

What does the modal particle "auch" add to this exclamation?

The meaning of this exclamation is something like: "Make sure you behave!"

Auch reinforces commands.  This is similar to the English “Be sure” or “Make sure…!”

Unser Urlaub in den Staaten war einfach toll.

What does the modal particle "einfach" add to this statement?

The meaning of this statement is something like: "Our holidays in the States were simply great. (There's no other way to see it!!!)"

"einfach" used as a modal particle emphasizes that alternative possibilities are excluded

Warum hast du denn nicht gesagt, dass es schon so spät ist?

What does the modal particle "denn" add to this question?

Here the meaning of the sentence is something like: "Why on earth didn't you say that it's this late?"

If there is a negative element in the question, as here, "denn" signals reproach.

Warum ist sie nur so unfreundlich?

What does the modal particle "nur" add to this statement?

The meaning of this statement is something like: "Why on earth is she so unfriendly?"

"nur" intensifies w-questions and can make them sound more urgent.  In this case, it makes "Why...?" into something like "Why on earth...?"

Was kann man auch dazu sagen?

What does the modal particle "auch" add to this question?

The meaning of this question is something like: "Well, what can you say to that?"

"Auch" can make w-questions into rhetorical questions (expressed to make a point and not because one expects an answer)

Was macht sie eigentlich?

What does the modal particle "eigentlich" add to this question?

The meaning of this statement is something like: "What is she really doing? (Come on!  I know you're not telling me the whole story!)"

In w-questions, "eigentlich" implies that the question has not yet been answered fully or satisfactorily.

Was war das auch für ein Erfolg?

The meaning of this question is something like: "Well what the hell kind of success do you call that?!"

Auch with exclamations emphasizes the speaker’s negative attitude, in this case rendered with "what the hell".

Wie kommt man denn am besten zum Schwimmbad?

What does the modal particle "denn" add to this question?

Assuming that the tone of voice is neutral, the meaning of this statement is something like: "Tell me, what's the best way to go to the swimming pool?"

As is the case here, "denn" most often serves to tone questions down and make them sound less blunt and more obliging

Wie war Ihr Name doch? (It could also have been: "Wie war doch Ihr Name?")

What does the modal particle "doch" add to this question?

Here the meaning of the question is something like: "What did you say your name was again?"

In w-questions, "doch" asks for confirmation of an answer or the repetition of information.

Wo bist du denn so lange geblieben?

What does "denn" add to this question?

This question correponds to something like: "Where have you been all this time? What on earth have you been doing?"

"denn" can signal reproach if there's a negative element to a question, as in this one.