Is it possible to have a cousin that has the same last name as you?

Of course.

yes, of course. your father's family may have many people with the same last name as you.

Yes

Only if you marry them

Yes, on paternal sides.

Yes. If your father's brother has children, they will have the same last name as yours.

Yes it is

It's almost certain to happen as it more of a question of your family tree & if your parents have any siblings. You generally have to look at your father's family (as most families go with the father's last name) for brothers that have families as well.

If you mother didn't marry your father (or your father elected to take your mother's last name), you will retain your mother's last name & then will need to check for your mother's brothers for families for occurrences.

Unless your parents are single children, there's a fair probability of this occurring in the immediate family tree... you can extrapolate this by generations & the probability can increase even further.

Yes, if the 2 fathers were brothers, their children would be cousins and have the same last names.

Certainly.