Forbo makes stair nosing for their Marmoleum.

I did our basement stairway same as in the pic. It's basically a wall built adjacent to the stairs, then attached to the side of the stairway and floor. It's not hard to do. With a fully enclosed wall, you probably have a floor-to-ceiling post at the bottom, which would need to be removed, and then you would likely have to attach to the stairs instead. You can start by removing the drywall and possibly just making adjustments to the existing wall.

I don't think you want to "start" with flooring...I think you want to start with a plan for the whole room. That way when and as you buy the pieces and materials and start putting it together, you know you are continually moving toward the desired final outcome. So start with function...how many people do you want to seat in the room? How will the room be used besides a morning cup of coffee? Think about traffic patterns as it appears there's one from the stairs, one from the kitchen, one from the front door. You want to make sure your furniture arrangement does not obstruct the flow. Think about lighting...where are the outlets? Do you have any floor outlets? If you know your furniture arrangement, you can have them added before new flooring is done. That way you don't have to run wires across the floor which are trip hazards or under carpets which are fire hazards and illegal. Do you want some ceiling fixtures? Do you want to add lighting to the bookcases? Do you want to add some crown molding to the room to dress it up? Also on flooring, it appears there is some kind of transition material at the front door...do you want to unite that with the rest of the room? Where can you start and stop the new flooring? On some specifics, even if you don't want to add the bridge to the bookcases, which would really make them look nice, think about separating them with a sofa in between. It will help make that wall a focal point. For window treatment, I would think that, with the dog, you'll have to do some kind of shade that drops down from above...either duettes or roller shades or roman shades. The idea being that if you lower the shades by 2/3rds at night, you'll still gain a lot of privacy but leave room for the dog to look out. I don't think a single shade is doable as it would be so wide, so probably just divide the window up evenly and have the shades butt at close as possible. Depending on the style you want in the room, you can add a valance or side panels or both. I'm not a real fan of the railing by the stairs as it looks insubstantial. I'd think about maybe replacing it with something else...maybe metal spindles with a wooden top and a good newel post. That you'd want done before the flooring if you want to do it. On the chairs, it is hard to find new chairs that are comfortable and smaller. I wanted a new chair for my bedroom that was smaller, so I ended up buying one off of craig's list and reupholstering it as I couldn't get them that small new. Since you have them and like them, then leave them if the fabric is in good contidion, or redo them or think about getting the slipcovered which will save some $. And I'm sorry, but I'm really not a fan of the large wreath on that wall. Art on the wall is one of the last things that go in a room. Right now, I think it's in the way of your seeing the whole picture of how the room could look (perhaps your reluctance to separate the bookcases even further?) I'm going to recommend you watch Christopher Lowell's 7 Layers of Design as it's a nice way to get you thinking about designing your room with consideration to all the elements as well as see how they all come together to create a final room. And as he always says, You Can Do It!

I just did LVP on my second floor, changing out both old laminate and carpet. The main reason being my little shih tzu loves to ‘stealth pee’ upstairs and I rarely find it until the next day or so, especially when she sneaks in a room that we don’t use daily. I priced out both engineered wood and LVP and they were actually similar in price. I also LOVE that the vinyl is soundproof. My teenagers are quite noisy and I really wanted not to hear them like I did with my old laminate. As for carpet, we have no place for it anymore since we can’t trust my doggy. Lol We did a medium grayish-brown that has texture. I really love it.