A well-lit photo of a room is tremendously helpful. The Pros on Houzz will almost always ask for one. Nonetheless, I will give you my personal thoughts on design with color. To me, white, black and most woods are neutrals. Gray is a combination of black and white, and therefore another neutral. However, gray can also have a tint or shade of color. Most gray is bluish, but any color can be added to gray. I prefer the woods in a room to match—the same color or wood type—but they don't have to. With strong colors it is important to spread them around the room. The photo below does this very well. Dark blue and white wing chairs are balanced by dark blue and white lamps either side of the sofa. The light blue accent above the French doors is balanced in accent chair, ottoman and throw pillows. The red upholstered coffee table anchors all the other red accents. The problem with adding new fabrics to an existing color scheme is matching the colors. The lighting in a store (on online) is different than the lighting in your home, so it's best to take something with the color you want to match to the store. If you want to modernize your room, it would be better to get gray slipcovers.

I think a black velvet sectional sounds nice. Is there a television on the fourth wall that isn't pictured? or what does the couch currently face? Also, a nice area rug could be way to disguise the carpet and ground the space!

Love the artwork and I would get a sofa and two chairs also. I love the blue in the art, maybe use that tone for the sofa and chairs. This would look great. https://www.westelm.com/products/haven-loft-sofa-86-h4998/?catalogId=71&sku=6929507&cm_ven=PLA&cm_cat=Google&cm_pla=Furniture%20%3E%20Sofas%20%2B%20Loveseats&region_id=658780&cm_ite=6929507&gclid=CjwKCAjwlrqHBhByEiwAnLmYULTzSF_1EJIWPUjqxZM7mtGNOD3tBLJbV4rF-UD6W6hk88TRTe-nshoCEewQAvD_BwE Maybe you can find one with microfiber, which is the best product to use with children.

Almost anything goes with white walls. I think the problem is you don't really know which style you want. You named three in your question. It's not so important to be able to name it, but having some inspiration pictures so that you can convey the look you're after would help. If you haven't already, take some time to compile one or more Ideabooks on Houzz (I do one for each room). Just start saving pictures of rooms you like and indicate what you like about each room in the comment field. After a while some commonalities should start becoming apparent. If not, if you're still all over the place, understand that sometimes one house or one room can't hold all of your good ideas. You will have a more pleasing, coherent end result if you refine your taste. Save some ideas for the next house, or the next time you redecorate. :-) What makes professional decorators'/designers' rooms stand out from non-professionals is the discipline required to achieve an intentional look.