What I'm worried about is the stairs. I can't see where they would be
able to get up or down at all. Some of the options I've considered:

- carpet the stairs - to me this would seem odd because it would be the
only carpet remaining on the top 2 floors of the house
- laminate the stairs and install a "runner" down the middle 12 inches of
the stairs
- laminate the stairs and find a "safety tread" to install on the
bullnose or the step of each stair

Consider either full stair carpeting, or a runner, english style, with stair
rods and brackets to hold the runner against the bases of the step risers.
I would use something wider than 12 inches, perhaps 18 inches, so humans
could use it too, and it would look more balanced, covering 2/3 of the width
of the stairs. Agree with person noting that your dog's toenails should be
kept trimmed so most of the weight is on their pads, for better grip. On
our wood stairs we have stick-down carpet rectangles that are glued to the
wood. The dog gets good grip, but it looks like hell.
They also sell rubber tread covers, but they look kinda institutional, but
help animals get a grip, as it were.