As Milly and Joseph said, you would have to open up a few places to see your framing details etc, I mean if you have experience and knowledge you would pretty much know how the framing is and what is being carried by what and you will have to make an opening, but existing joists, double up some existing, pretty much like you would make a pulldown downstairs, the concept is the same. Stairs can be built onsite depending on the staircase you want to have if it's just a boxed set of stairs between the 2 walls, it's very easy to make them onsite, again have to know carpentry. In any case, you will need an architect, a set of drawings, building permits, etc but this is pretty much what needs to be done just to give you an idea. If you think you can do it, nothing wrong with it to try or at least you can do what you can and hire someone to do what you can't do. I remember in my old house, a couple a few doors down build a 2 story addition, they had some family members helping them lift the walls, etc and they hired a contractor to frame the roof and do the sheeting. The guy I think owned a car parts store and the wife was a nurse. Little by little they did everything. Good luck.

I think you would be making a very big mistake. Your stairway would not look good with tile - even wood-look tile. If one uses that, it really should be done at the time of construction to make it work properly. I can't imagine why anyone would want tile on a stairway unless it was a Spanish/Mediterranean house and then the tile would be red and look like tile.

I might be unusual but I probably wouldn’t try to match them. Id do a contrast instead. Maybe go super light or a very different tone on the basement floor. something that harmonizes but is definitely different. Good luck whatever you decide!

Food for thought - If you are going to the trouble to get new hardwood floors (and I agree with you not wanting LVP), then I would get what I want as it is a long term investment. If you like darker rich floors, I'd get them. Down the road when you want to paint or change out your kitchen cabinets or sand and stain the stairs you can do so. It will be a phased approach. For example, I redid my master bedroom/bath and redid the floors in dark walnut because I love the warmth. My other wood floors in the house look like the color you have. I've lived with them and this summer am going to refinish the rest of the upstairs. Then I'll move onto the downstairs. It's better to do what you want/love instead of picking something to "match" something that you don't really like. One thing you could do (if you don't have an objection) is to paint out your stair spindles and risers white (that's the way many homes are ) and leave the railing and stair treads as they are for now until you refinish to match the new hardwoods. You could also paint out the interior trim/doors/woodwork white as well. It will significantly lighten the space.