If you can remove the popcorn ceiling, it will make the ceilings look cleaner and newer. Since you have children, you need to be careful to not generate dust, to keep them from developing allergies or asthma.

https://www.google.com/amp/s/pulse.seattlechildrens.org/the-dangers-of-home-renovation-keep-kids-safe-during-repairs/amp/

Caution: some popcorns ceilings contained asbestos.

https://www.google.com/amp/s/www.asbestos.com/blog/2018/07/24/asbestos-popcorn-ceiling-safe/amp/

The walls will have a better finish if you remove the wallpaper, scrub the walls to remove the adhesive (a wet roller mop works wonders) and then spray new texture to cover over where the adhesive was removed. Prime the walls and paint them in white flat paint. Then, using the same shade of white in a semi-gloss finish, put an extra coat of white on the woodwork (baseboards and doors).

The screen between the dining room and hallway is non-structural. You can remove it but would have to patch the floor, ceiling and connection at the sidewall.

Do everything white. If you don’t like the hallway, you can always find a way to rework it later with doors or cabinets.

Sherwin Williams has 50 shades of white. You should be able to find a shade of white that is perfect for your home. They have a lightbox in the store that enables you to see the paint colors under different lighting conditions. Right now, they are having a 30% off sale: https://www.sherwin-williams.com/homeowners/special-offers/digital-media

Wow, that wood screen is gorgeous. In line with the advice to pick one color and go with it, I just painted our entire house one very light grey throughout. Figure it will give me a chance to experience how light works in the various rooms, especially as we have to bring down some trees (for safety), so that will change the light pretty dramatically.

A good way to pick a paint color, including white, is to coordinate with the semi-permanent hard surfaces, aka, the flooring tile and fireplace. If your floor tile wasn't cleaned professionally before you moved in, try getting that done first, as it will lighten up considerably and look like a different color. Trying to coordinate with the old light green is probably limiting your choices, so I'd recommend painting it white sooner or later.

For good contrast with the shadows of the fretwork, choose a light color. muted wallpaper for the hallway. New wallpaper with a vertical pattern will disguise the secret storage doors.

I would not choose a paint color on the basis of what will best camouflage the storage door/s. It is what it is. That said, it's not necessarily weird to paint one wall a different color, as long as you take into account the rest of the space. That wood screen is worthy of focus, so I would try not to do anything that would detract from it. No wallpaper, no accent wall.

Behr's COTTON GREY is an almost white color that would look great with the wood feature wall. I would use it in all of the connecting rooms with a white trim and white ceilings. Double check, but I think it will go with your tile floors, too.

I would take the plunge & paint all trim and doors. Neutralize them & now your color possibilities are endless. This is a complicated one & it might be worth getting a designer on-site for a bit. Gl :)

THink of the white paint as a way to get a fresh clean basis to add color. If you are priming the walls with white primer, you'll get a good start.