Can I play Hasty Exit if I don't have control of the Battlefield? My first instinct is no, because I can't GIVE control if I don't have control to begin with.

That should be correct. Anything with ‘to do this’ or ‘then’ requires that the first condition is met for the effect to occur. You wouldn’t be able to meet the condition of giving the battlefield control if you didn’t already have it.

You could play it, but unless you control the battlefield, it won't do anything.

It is like spotting a character.