I'm planning on floating an engineered floor in my living room over the current 3/4 red oak (lots of water stains, easier than refinishing). There are 5 oak stairs leading up to the room. Do I glue/nail the engineered flooring to the oak treads of the stairs? What about the nosing on the stairs - cut it flush with the riser? Also, along each side of the staircase there's a thick board that runs up the slope of the stairs (sorry, I don't know what it's called). The treads & risers butt right against it. It's been painted an ugly dark brown, and it's pretty roughed up. Is there a way to cover that?
Edit:
What was I thinking? OF COURSE, there's a wonderful tutorial on how to cover stairs. I LOVE this site! I still have the question of what to do with the skirting - that's the piece I couldn't name. It's been banged up over the years by vacuum cleaners, kids, etc. How can I cover this?

I think you mean the skirtboards. You could either replace them or cover them with 1/4" oak veneer plywood stained to match and cover the top with an oak base cap.

Thanks, Jerry. That oak veneer sounds like a good idea.