If I have dry eyes, does that mean that I have damaged my eyes by using my contacts?

No. Tears are needed to bring nutrition to the cornea and carry away the waste products. The most common first problem is pain when not lubricated, but if one tolerates the pain long enough one can damage the eye.
No. No. It just means that the contacts are rubbing on the eye due to the decreased tear film and dry eye.
Usually not. Dry eye symptoms can go on for a long time before truly permanent damage occurs to the cornea like scarring. If your contacts are uncomfortable this may mean anything from dryness to fit. It's best to be examined and treated rather than letting the problem persist.