Differences between 「が」、「ても」、「のに」

Question

What is the usage difference between these three definitively? As an intermediate Japanese speaker, I somewhat understand where I would say them, but I can't point out under which contexts any of them "wouldn't be allowed" compared of any of the others. In speech, I have no problem selecting one for usage, but I just can't pick out if, when somebody else uses one, they are using it in an "incorrect" way (I can figure out if it's natural or not).

〜が is largely a "but", where you give a contradicting statement afterwards

〜ても is closest to an "even though" suffix to a word

〜のに is a more emotional "が", with a connotation that even though an action was taken, that something has / hasn't happened.

Where is each NOT allowed under the other contexts?

Answer 1

が

 ○ 私は電車で行きますが、あなたはどうしますか。
 × 私は電車で行っても、あなたはどうしますか。
 × 私は電車で行くのに、あなたはどうしますか。

The spirit of が is not contradiction, but that the next phrase comes from "other direction". Since English language doesn't have a word wholly translates it, it could become "and", "while", "when", or even "so". But as far as I know, Korean, Vietnamese and Polish have similar notions.

The colloquial equivalent is けど.

ても

 × 何度食べたが、ブロッコリーは嫌いだ。
 ○ 何度食べても、ブロッコリーは嫌いだ。
 × 何度食べたのに、ブロッコリーは嫌いだ。
(○ 何度も食べた[が/のに]、ブロッコリーは嫌いだ。)

As you said correctly, this word corresponds to "even if", "no matter", or "-ever" in whatever, whenever. Thus, only this one is grammatical with bare question words.

のに

 × 約束したが、どうして来なかったんですか？
 × 約束しても、どうして来なかったんですか？
 ○ 約束したのに、どうして来なかったんですか？
(○ どうして、約束[したのに/しても]来なかったんですか。)
(△ 約束しましたが、どうして来なかったんですか？)

This one is a loaded word, that conveys the latter statement contradicts with expectation or obligation the former one implies. In most cases, it's almost ungrammatical to use other words when the latter clause carries accusatory connotation.