Yes, the hard drive is shot. That's not surprising after a drop. Replacing it yourself is dead easy — instructions are in the little owner's guide that came with it — and if you do so, you will get your choice of drives, a lower price, and a much longer manufacturer's warranty on the new drive than Apple would give you.
Your warranty on the rest of the computer is gone, though, as a result of the drop. The dent in the bottom may be purely cosmetic or it may not, but in either case it is +prima facie+ evidence that the MBP has been mistreated, and that invalidates the warranty. If you bought AppleCare extended coverage in addition to the one-year warranty, that is also void now, and you may want to call AppleCare, explain the situation, and see whether you can get a pro-rated refund of part of the cost of the AppleCare plan. Some other people have reported success doing that.
If, after replacing the hard drive, you find that the machine still doesn't work completely right, you may want to bite the bullet and pay Apple to restore it to like-new condition. I believe doing so would restore your warranty and AppleCare coverage, if time remains on them.