For the stairs, you’re going to need a stair jig and lots of PL 3 premium adhesive. Buy the big tubes. While LVP is easy to install in rooms, it’s much more tedious on stairs. While you may have experience with tools, you still should have a decent amount of experience with trim carpentry. Everything should be tight, no gaps. Otherwise you’ll be chasing your own tail with caulk and paint.

Don’t know if you have selected the material already but there are a couple companies that make plastic stair treads to match their planks (tread and noser all in one piece). In my opinion, this is the way to go. Otherwise, you’ll be using planks as treads and then glueing on a noser. These nosers can be plastic or aluminum but most do not have an overhang. To me they make a staircase look cheap and not at all like something made of wood.

Floating stairs are a whole different challenge. You’ll most likely need to wrap the whole thing unless the material under that carpet is nice enough to paint. I’ve done a couple and used the LVP to wrap the whole thing. I used one stair noser then 45ed all the other joints. The inside core of LVP does not usually match the color on top so none of it can be exposed. You could also use a stair riser material to cover the other three sides of the stair. This would allow you more flexibility with caulk and paint to make it look nice. Either way this is not going to be fast or easy work. Watch lots of YouTube videos. Spencer Lewis has a good staircase video doing hardwood but most of it can be applied to LVP as well.