If I give control of the battlefield to my opponent that is not the same as them claiming right? Next can I still claim it back that round?

Correct on both. Claiming gives you control of the battlefield, but it's not the only way it can change hands.

Correct. Hasty Exit is just control. They don't get to use the claim ability via YOUR hasty exit.

And you can still claim that round right?

Yes you can still claim.

Do you actually have to have control of the battlefield to play Hasty Exit?

It does require you to give control of the battlefield to remove a die, but I guess you could play it and not give control to not remove a die if you wanted to.

Control of the battlefield is a game status that changes based on who claims the battlefield. The player that brought the battlefield chosen for the game starts out in control, and claiming the battlefield also gives you control of it. You keep control until your opponent claims, or a game effect changes control. Changing control is not the same as claiming: It is an effect of claiming.

"To" effects are similar to "then" effects. In order to get to the part after the to/then, you need to do the step right before the to/then. So if you don't give control of the battlefield under Hasty Exit, you don't get to do the part after the "to." So no dice removal.