I'd post a current floorplan/layout in the Remodeling forum with your wants and needs, and get suggestions there. This forum is more about building (or re-building) porches, not demolishing them. :-)

Sorry...but you're using baseboard, so a skirt board would have been so much better and more consistent. When will you install the bullnosing on the stair treads?

Traditionally you 'match' the floor that is in the public areas...that is to say the main living/entertainment of the home. If everyone and their dog hangs out in the basement and you entertain in the basement and you serve formal or semiformal dinners in the basement, then match the basement flooring. If you have a traditional house with a stair case that is visible (open...not a closed door at the top of the stairs) from the main living space then you match the stairs to the main living space. Basement flooring can be stand-alone.

Copy that! You may be able to keep what you have in place, but I would certainly make sure it has dimmer control so you're not over-bright in laundry. One additional idea for this laundry space would be undercabinet lighting (so simple now with LED strips or pucks) that's also dimmable. That might give you more targeted task lighting while working out a stain or soaking items in your utility sink without needing to reach for the overhead light. Alternatively, it could be helpful to consider moving the switched box location from overhead to the far wall (rt side, from in front of laundry units) that could help provide a beam angle into the laundry units over-shoulder for sorting and transfer, since there's less wall surface behind from that perspective that isn't shared with the hall area. In tall, narrow stair halls like this, I often recommend a plan of wall lamps that uplight the tall vault and bounce the light down into the pedestrian area versus directional beams down from overhead. When placed at the right elevation over and above personal height, they don't have to interfere with traffic clearances even in a super-tight space. Hope that helps, cheers!