Correctness of questions without inversion, relying on intonation

I hear and read them all the time. I mean stuff like:

You're just going to stay here?

Instead of:

Are you just going to stay here?

Then I write like this out of habit and get called out on my grammar issues by people who speak English much worse than I do. That's kinda embarrassing. So, could someone please explain to me whether it's correct or not? Are they acceptable at all, and if so, in which situations?

So, what? You're gonna kill me now?

I believe @BillFranke has answered this appropriately.

It matters a great deal whether or not you are using this in conversation or in a writing - (let me make the distinction that I believe a written Conversation still falls under Conversation rather than writing - perhaps Authoring would be more appropriate a word.

I ran into this problem a great deal when I first started writing. There were several Authors that had been rather outspoken about specific rules in writing, for example; Steven King in On Writing states that the use of Adverbs is unnecessary and should be eliminated from any final draft you create. I've also heard "...never start a Sentence with "And!"

As such, I found that often times, without thought, I would begin a new sentence AND paragraph with the word And. It drove me absolutely nuts that I had such a predilection towards, what I consider, the improper use of the word and. Then I realized why it was so common place in my interior dialogue...

It is used constantly in conversation. For example, in the work environment, you might hear someone say, "Printing out these reports when we save them digitally is ridiculous." and while that may seem like the end of their sentence their mind will conjure up another point and stress it with the word and, such as, "And, its not efficient either."

As Bill Franke put it, it is more a sense of style than a rule of grammar, after all, we often don't speak how we write, though we always try to write as we speak to keep the conversation as authentic in verse as it is in reality.

It's perfectly grammatical. Whether or not it's appropriate depends on what you're writing; in some contexts certain constructs are considered inappropriate: for instance, if you're writing a formal business document it'd be wrong to write you're or it'd, but in an answer on StackExchange those are both fine. Those kind of restrictions attach to vocabulary as well as syntax. So people are wrong to tell you that these questions are ungrammatical.

As a degenerate case of sorts, if someone claims that a grammatical rule restricts questions to sentences with either the particular question structure (Is it raining?) or particular question words (Who is that?), ask them what the 'correct' equivalent should be for one-word questions (Me?), which are surely acceptable.