I think instead of conventional bookcases, I would do some iron and wood shelving with the wood being natural -- not white. I think a sofa table in the same hue as your dining room table would look good behind the sofa.

Do not put bookcases there .... there is not enough space to do that. Book cases are for large walls with no windows.

What pleasant rooms! I'm no expert so can't help you, but it looks very nice.

I think I would do white. It would keep the room light with all the dark furniture. Are your kitchen cabinets white?

I think you have enough room. Most bookcases are not that deep and even if the lower portion is 24 inches, I don't think it will crowd out the room.

As far as the sofa table, find a piece you love. It could be a painted piece in the soft tones of you accent colors, metal, or a mixed woods. Take a look at console tables in products.

I think shelving that wraps the inside corners and has some of the same detail as your built ins (if white) or your dining table wood tones if in wood. The home is beautiful, I think something unexpected would add some pizazz to that game area, for example if the shelves wrapped the corner and ended at different lengths, or had corbels supporting them. Shelving will let you change out the decor and do seasonal things and add some character. I'd like to see a purpose driven use of the space as well, for example a chair and ottoman in the corner with a side table. (Always sucks to be the first one outta the game!)

Start with drapes to warm the space. When hung properly, there won't be room for bookcases. In fact, most Houzzers would advise you to limit the amount of photos to just a few. I like them with "foot" backs so they can sit on table tops. Group them in threes, one large behind two smaller ones; no more than one such grouping per table top. The rest can hang in hallways and bedrooms, or go in albums. As for things hung on the walls, they should center at 5' above the floor. If above furniture, then just 5" above. I think your kitchen table and chairs are fine. Is that a sample valance over the slider?

For softness and sound dampening, I would prefer drapes, but fabric roman shades work well too, and can match the drapes. They can be inside mounted so your nice woodwork is exposed. Valances are not popular on Houzz - except in the occasional kitchen/dining combo.