+1. I hold on to the anchor power point with my left hand and hold onto the loose rope with my right keeping in brake position. I then slowly let my right arm out until it's locked out. Then a little back step to tighten up everything. I'm especially looking for the anchor lines to be equally stressed and will side step accordingly if necessary.
I didn't think thru all of your description of the anchor, but basically if you are tied off to 2 solid trees you are in great shape.
1. make sure the TR biners that will hold your rope are dangling over the edge. do not lay them flat on the surface near the trees. your TR rope should not run over the top edge of the rock - that is a bad set-up.
2. as a consequence of (1), you will have static line (or slings) going over the edge to the biners where the rope feeds. pad the slings or rope if the edge is sharp - use some old T-shirt. the padding should protect the slings/rope, but not interfere with the rope feed.
3. NOW for your real question. Yeah ... remember that well. It always felt like a bit of a bitch to lower yourself over the edge and get your weight on the rappel system. Remember this ... HOLD ONTO that brake hand and NEVER let go with your brake hand. Gently work your body over the edge, and there is a small drop of 6-12 inches before your weight goes onto the rappel device. usually I just muscled this - holding the anchor system with my free hand.
Again ... you are fine so long as you never LET GO of the brake hand. People only get in trouble when they panic and release that brake hand.
then you are off and rapping.
P.S. I do not like autoblocs and prussiks. others disagree. make up your own mind when you have more exprience.