Can a future employer ask previous employer if you gave 2 week notice and if you are eligible for rehire?

Yes, of course. A prospective employer can ask anything he/she wants to ask as long as it isn't discriminatory - it's whether the previous employer wants to answer that matters.

They can ask a great many things. The ex employer may not give any answer at all to avoid possible problems. Whatever they do say must be truthful. Why do you ask?

Yes, that is just the tip of what they can ask, were you on time, did you do a good job and more

Yes they can ask. Whatever the question, just be certain to answer every question honestly, as they do have ways to find out if you have told them the truth.