Do objects in Earth's orbit appear to be weightless because they are in free fall?

Yes, exactly. A useful way to think of an object in orbit is that it is falling towards Earth much like anything else but, because of its velocity and distance from earth, it keeps missing the ground. This seems slightly contradictory -- a weightless object under the pull of gravity -- but it isn't, really. Suppose you're in an elevator that's had its cables cut. (Heaven, forefend!). Gravity accelerates you and the elevator Earthward at exactly the same rate, so you feel like you're floating relative to the elevator, but you and the elevator are just falling at the same rate.