KInda depends on how much the stairs move or squeak. If they are noisy or move quite a bit i would fix that. But if they feel solid i would just lay right over them. Also if the stringers for your stairs are in good shape you could just replace the treads and risers and that would be quick.
Yes that is the terminology I was looking for, haha. I will have to pull it apart to see what things look like. They squeek a lot, probably should just all be replaced. From what I have looked at, they are very hacked together.
