"I'll get mine" or, "I've got it" is a casual way to say it in American English

Yes, but it means that I am able to pay - I have the money to do so. That is not quite the same as saying "I WILL pay for mine', which is a statement of intent.

Yes, it is fine. You have many options: I can pay for mine. I can pay for my meal. I can pay for myself. I can pay for my own. All of the above are correct in this situation.

yes, in context, that would be an acceptable sentence.

Yes, that's perfectly correct.