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I would definitely paint out timber frames, they in themselves become the feature if you don't…..and I would hope you mean a soft grey! To "hide" skirts and arc's (I renovate and sell for a living) I do them the same colour as the walls but in a satin finish…..you could do semi gloss….a water based enamel works best. Make sure you prep them well though, either a good sand to rough up, or something like ESP. I always thought rule of thumb for a feature wall, paper or paint, is the wall opposite the greatest light source, ie. the wall that gets hit with the most natural light. Failing that, I think it would look lovely on the alcove/bay window wall and do the inset of the alcove/bay window in a contrast….say a soft version of the pinkish hues in the paper??? You seem to have some substantial pieces of furniture, and a piano, so doing the other walls would compete and look overbearing to me. Unless you are going the whole Regency style embossed paper/stripes/pattern everywhere…………...

Can you provide a picture of where you're planning to put the paper vs the paint? I think, myself, that wallpaper should be continuous along the length of the wall, unless there are architectural features that break up the line of sight.

As I said before keeping clutter under control is the key, get lots of storage ,you said cost is I s an issue. Look at IKEA they have solutions for wall units that you can get in different colours and that in itself will brighten the place. You would benefit from a buffet next to the kitchen /dining area or a large wall unit with lots of shelfs and drawers, coffee table with storage drawers , storage unit with baskets for kids toys and school stuff, once you have this under control start adding rugs if think you still need it. By the look of it you have small kids get them to put things in appropriate place and get some funky coat hangers in the entrance so they can hang hats and bags etc.