I'm confused about these two sentences:

Are you staying for an extra week? And Are you staying an extra week?

I know the for is unnecessary but it's also not grammatically incorrect to include it so what's the deal here? Should I use the preposition or not and is it considered too formal in everyday speech if I do decide to include it?

Thank you so much.
I would consider both expressions register neutral. That is, both of them could be used formally as well as casually.

The only possible difference I see is that without the "for," we're more likely to assume that the stay is planned for in weekly increments.

That is, if it were a monthly rental, for example, and you arranged to extend it by one week, the "for" would be more appropriate.
Students: We have free audio pronunciation exercises.
But don't both mean an increment of an extra week?
Yes, of course. The difference is slight. I'm suggesting that in one case you're adding the week to other weeks, while in the other case you may be adding it to a month.
What if it were only an extra day being added to a two-month stay?

I tried to make it clear that I'm not suggesting that the "for" makes an important difference.
I'm just saying that in a particular situation you might be more likely to use "for."