Foot down?

[With_Two_Flakes]

In other words, if the receiver is along the back line of the endzone and has his heels touching the ground but not his toes (say he's falling backwards) is that enough?

Yes that's enough. Everyone sees the NFL and top NCAA players do awesome catches where they drag their toes inbounds to make fabulous catches. Dont so often see your scenario of someone dragging a heel, but that would be no different.

If the reciever is on the sideline and he drags a heel and then his toes hit OOB is that a catch?

That is more difficult. If it is toe,heel or heel,toe then no catch - the philosophy in this situation is to regard the foot as one item.
If it is toe......heel or heel.....toe with a definite and distinct period between the two, then it is a catch.