Well, in another screenshot (which I am too lazy to look up) you can see the other side of one of these things, and they are covered in targets, with little benches and buckets of arrows in front. So they obviously must be art exhibits ...

Beats me why it appears to be pained, but I'd expect it's supposed to be strong enough to stop the arrows.

Well, you'll notice that the cloths are facing the OUTSIDE of the city.... maybe they're only to hide what's going on from enemies.....

Maybe it moves to provide a visual distraction for the archers. Those supports could be rollers........ although that sort of stretches it even more than it is already stretched.

What concerns me more in the field of pointless minutae is that LADDER going up to the top of thecloth (or maybe the tree):

What could this be for? It's not on the other "stage", and it seems to connect to some kind of platform on the tree.... or are there different kinds of "stages"?

Yeah, that platform is what it made it look like old movie stages.  If it were any sort of theatre stage...    well 1st off, it wouldn't make much sense because at least I never heard of a backdrop being used in a play prior to the industrial revolution.  If it were an outdoor stage, you'd think it would be given some sort of walls or ampitheatre setup so that the audiance could actually 'hear' the plays being proformed.   And keeping a backdrop like that up for a play would be a nightmare unless the channeler was keeping the wind steady (or the cataclysm destroyed wind and the channeler has not channeled it  back into existance yet).   For archery ranges, you'd only put up the cloth (assuming you'd put up the cloth at all) when you're practicing, which you would not want to do on a windy day anyway.

Perhaps those  are the gallows, and they like to hang people with a happy landscape background so that those being put to death have "a happy background" to their deaths.