Don't think this really needs to go via exile; but ok - everyone exchanges hand and battlefield.

I think you probably want to add 'non-land' to the removing from play though - otherwise this will lead to very frustrating games.

How would I word it without exile?

Just that. Something like: All players put all permanent cards from their hand into play, and return all non-land permanents they own currently on the battlefield to their hands. (also 'own' versus 'control' is kinda important there)

I'm not a good person to ask about templating though; it's possible the ugly exile zone is needed.

I think that that wording would result in all of them ending up in the hand because those put onto the battlefield would then be affected by the second part, but not sure.

I think it's cleaner to go through exile. Otherwise you'd need the word "simultaneously" in there (because otherwise spell effects happen step-by-step). Compare Living Death.

That was my formatting precedent, too. My other worry with this is the cost. Is this overcosted?

It has the potential to be a crazy combo piece, so I don't think it's over-expensive. It doesn't quite work because the hand is a hidden zone; you have to add a clause about revealing the hand before players exile specific cards from it, unless you want it to be optional.