If an object is, say, 100 cm. from a wall, and I move the object halfway to the wall and stop, then the distance is reduced to 50 cm. If I continually move the object by one half of the remaining distance and stop, and keep repeating this procedure, why is it that the object eventually makes contact with the wall (on a macroscopic level at least), considering that no matter how infinitely small the remaining distance is, I can theoretically cut that distance by half, and thus never reach the wall. Does the Planck distance, the theoretically smallest unit of distance, have anything to do with this? And if so, how does any object get "around" that infinitely small Planck distance in order to move at all?
The flaw in your setup (and you can just say "Zeno's paradox", it's ok) is that you think that the distance between two objects has to be zero for them to "make contact." "Making contact" is largely a matter of electrostatic repulsion between the two objects. At some point, they become close enough that their atoms push apart, and then they can't get any closer.
As for movement, Zeno's error is in assuming that the infinite number of steps would never complete. Modern mathematics is quite adept at dealing with limits and infinities. Since every step which is half as far also takes half as long to complete, the sum converges and movement is (as you may have seen) possible.
Finally, the Planck length is...really pretty much never the answer to a thought experiment. As Wiki says: "There is currently no proven physical significance of the Planck length."