Typically, you can ship your entries in at least one week before the judging begins. In larger competitions, that might be more like two or three weeks. In all the judging I've done, I've never had a problem with sediment getting into the entry. If you have a properly conditioned beer, you should only need a day or two, tops, for everything to settle back out.
You should be able to get entry forms from a competition website or by emailing the competition organizer. If you see a notice about a competition, there should be some sort of contact information there.
Yes, most competitions require only two bottles, but the larger ones usually require three. Usually, you can enter two bottles in a three bottle competition, but you waive your chances as Best of Show.
Basically, pack the bottles well, get all your paperwork in the box (and on the bottles, as mentioned above), get your check easy to find with your paperwork, cross your fingers, say a prayer, hope for the best, and wait (patiently!) to hear the results.