Are そういう and そんな completely interchangeable? Is there any difference in nuance or meaning, or contexts where one is more appropriate than the other?
Most of the time they are interchangeable because their meanings are almost identical. The only notable difference is that in cases where the speaker is expressing negative feelings and wants to emphasize them, 「そんな」 is more appropriate, because it is more emphatic than 「そういう」. For example:
In those sentences, if you replaced そんな with そういう, they would be less emphatic. Of course, sometimes this might not be true because you can make them sound aggressive purely through intonation, however, it wouldn't "ring" the same.
I don't think the accepted answer is accurate, so I am writing a separate answer to address some differences I see between the two phrases.
There is an excellent answer that discusses 「そういう + Person 」. Per that answer, there are contexts where you can use either そういう or そんな. And this usage occurs in both positive and negative situations. For details see #2 and 3 in that answer.
But I think the statement that the accepted answer makes — "Most of the time they are interchangeable" — is inaccurate and too sweeping. Here are some situations and usages where you see the two phrases diverge if not completely differ.
When in a conversation one party, let's say A, says something and the other party wants to characterize A using the things A said or the way A said those things, you can only use そういう, and in this case, そういう means something to the effect of "you, who is saying this and that." This usage is discussed in the aforementioned answer (usage #1), but the example given seems a bit extreme, so I am going to give some more moderate examples:
B: うるせえな！そういうお前はどうだったんだよ！(Huh? You are telling me that? What about you? How did you do in the exam?)
I added 「うるせえな！」 to make B sound a little miffed, but one doesn't need to be angry to use 「そういうお前/あなた」. It could just be a joyful retort, poking fun at the other party and the irony of the situation. "Hey! You are telling me that? What about yourself?" In some contexts, it could even be as simple as a "What about you?"
A: いい年して彼女もいないの？！(You are not young. You still don't have a girlfriend?!)
B: そういうお前はどうなんだ？(Huh? What about you? implication: you yourself don't have a partner/lover either, so it is ironic for you to say something like that to me)
You can't use そんな in these contexts, because いう in this そういう is more literal than general, and comes from 言う.
Another situation where the two phrases are different is with the common phrases そういうことで/そういうことだから/そういうわけで/そういうわけだから (see this answer). If you substitute そんな for そういう in these phrases, the meaning changes and the phrase no longer serves the same purpose of wrapping up. そんなこと has more of a negative undertone than そういうこと. 「そんなことで怒る？」
Here is another one: そんいうことだ
「そんな！」 as an exclamation cannot be replaced with 「そういう」
Utterances that call for 「そんなに」 and 「そんな」 with に dropped also don't work with そういう.
Here, you can't use そういう, at least not without changing the meaning of the sentence.