You do not have to have a "sink", but you might be required to have a specific area to wash your hands and one to wash, one to rinse and one to sanitize your utensils and reusable dishes. I know you had mentioned putting a burner and propane tank on the trailer as well, great idea (for boiling water), you will need it for your dishwashing and sanitizing. KCBS rules (
http://www.kcbs.us/index.php) state (Rule 17) that each team must have a separate container for washing, rinsing and sanitizing of all utensils. This can be accomplished by simply bringing some jugs of water (5 gallon buckets, coolers with a spicket that can be turned on or off, not push button style, etc.) or have running water either on or in your trailer. Some even pre-boil all their water and put in a cooler, it will stay warm for quite a while, eliminating the need to boil or heat water on site. Of course for $250, you can install a propane fired, on demand water heater on your rig and not have to worry about heating water up. As far as the 4 sinks, you can either have a 4 sink setup, 3 together for washing utensils and 1 separate (available on ebay, under concession stand 3 compartment sink) for washing hands or you can buy 4 of the gray bus boy boxes from a restaurant supply store or Sams' carries them as well. Mark each one for what it's purpose is and you'll be in compliance. Of course quite a few teams travel in RV's making comps and cleaning quite a bit easier, but for those of us that don't have that luxury, lugging water and setting up cleaning areas is a chore. here in Texas, we mainly have IBCA contests and the rules are a bit more laid back. They do require you to handle all uncooked and cooked meats in a sanitary way, but I have never been checked for my conditions. Of course, I take great care with it all, because I'm feeding myself and my family with the leftovers anyway.
I have my 12 volt pump powered by one of the 12 volt rechargeable power supplys used for jumping off a battery. You can also just use a lawn and garden battery or wire it into your trucks power system.
As far as finding BBQ events, check the KCBS website under events for a complete listing (I think Tennessee has 15 events scheduled rightn now from April to October, but I'm sure there are a numer of others in and around the state to enter. Most KCBS events also have either a Pro Division (anyone can enter in) and a Backyard Division for those that consider themselves beginners or rookies.