Put 1" extruded polystyrene (XPS) on the foundation wall, then finish the rest as they require. The polyvinyl vapour barrier is pointless, but it won't damage anything, according to Building Science Corp. I've had to do the same nonsense on two new homes. The most economical thing to do along the stairs is to use 1/2" or 1" XPS, followed by drywall above the staircase. It shouldn't hang over the stringer. Underneath, you can go thicker, as it doesn't interfere with anything. Here is a link that might be useful: Basements

Depends on how wide your stripes are...if you want to limit it just to the landing area, I would do as many stripes on the right side of the doorway leading into the room with toys as you have done on the left side of the same doorway (from the trim to the corner). Then paint the rest of the stairway w/o stripes. At least that would "balance out" your doorway.

They weren't doing finish work before the roof was on they installed the stairs which are prefabricated before the roof was on so the framers could get upstairs to finish framing. I've been watching our builder's different house builds. In every case before the house was closed in, the subs got upstairs with a large wood ladder and not with prefinished stairs. And once the roof was on, they put in the bones of the staircase but non of the prefinished treads, etc. I talked to the foreman he said they will replace them which is fair to me. I'm glad to hear they're doing right by you but there's no reason for them to put them on until they're ready to lay floors, etc. So after drywall, etc.

I wouldn't/couldn't do it in my house, but is HAS been done. Here's an old post on it - [https://www.houzz.com/discussions/stain-floors-to-match-stairs-dsvw-vd~876622[(https://www.houzz.com/discussions/stain-floors-to-match-stairs-dsvw-vd~876622)