Is it OK to add a question mark to show inflection?

When asking a question you generally have to raise your voice at the end of the sentence, is it okay to stuff a question mark in order to show inflection?

A couple examples:

• 'That really happened?'
• 'I'm going to miss it again?'
• 'You did that?'
• etc...

I think you are referring to cases such as:

A: I'm so coming with you later!
B: Err... No?

In informal writing such as chat, it's perfectly acceptable, and other similar "stylistic" choices are fine.

In formal writing it should be absolutely avoided, since to express the same function there are other ways to achieve the same result in a better form.

All three of your examples are questions; they're just not worded precisely as such.

There is no problem at all in doing that (everyone does it, so it's good English), as long as the reader understands that the person speaking was asking a question -- so putting an interrogative point at the end clarifies that the person is asking for a reply, not simply making an exclamation.

I agree with what others have said about using the question mark to mark questions.

One aspect of the prosody of a query is whether or not it is a yes/no question.

Considering your examples:

'That really happened?' 'I'm going to miss it again?' 'You did that?'

These are are questions that may be answered "yes" or "no," and therefore have a rising inflection.

So would questions like:

Are you seriously considering that job? Do you want fries with that? Does this make me look fat?

(Word to the wise: never answer the last question with a "yes.")

Questions like:

What is your quest? What is your favorite color? What time is it? What is your name?

must end in a falling tone, or they will sound unnatural. (First two questions in this last list... see What's the connection between "Holy Grail" and "Killer Rabbit"?, if you dare.)

If you are writing, you should not put question marks on non-questions.

However, none of your examples are non-questions. All those question marks are appropriate.

The questions themselves are fragments, or else improperly executed, but that is forgivable in speech. As long as such questions remain quotations, you shouldn't have a problem grammatically.