Yes, of course. That is the only mechanism that can remove debris from orbit (apart from flying up there and shooting lasers at it and other untested fantastic ideas). And debris in higher orbits survives longer exactly because the earth's atmosphere gets thinner and thinner the further you are from the surface. The upper boundary of the atmosphere is somewhere above 100000km (62000 miles) http://en.wikipedia.org/wiki/Exosphere#Upper_boundary
Gravity doesn't do anything on its own, in the absence of other forces, the orbit of a steel nut around the earth would be stable (that's also why we haven't yet crashed into the sun even after being in its gravitational field for billions of years). (Well, you would get some instability because neither the earth nor the nut are perfect spheres or perfectly rigid, but those effects would take billions of years to become noticeable.)
"Gravity doesn't do anything on its own, in the absence of other forces, the orbit of a steel nut around the earth would be stable"
Actually, it wouldn't, because orbiting a celestial body radiates energy in form of gravitational waves, but in case of Earth's gravitational field and at the speed of orbiting satellites, this mode of losing energy is negligible.
Yes, but I still think that the tidal deformation of the earth under the gravitational influence of the steel nut will vastly dominate general relativistic effects like gravitational waves. You usually use neutron stars orbiting black holes to produce the latter.
Drag in space? At what altitude, more precisely, does the objects start to experiment drag or the lack of it therefore?