1) We use the tank & pump. If there is a leak, that annoying pump noise becomes a warning. In the middle of the night, if you hear the pump kick over, it's time to find the leak. With the silent operation on city water, you may wake up with the plywood subfloor soaked thru. Also, if there is a dripping faucet, it takes a surprisingly short period of time to fill the gray tank and then it can overflow in the shower (if it's the toilet valve leaking...even worse).
2) Both closed until ready to drain. Particularly the black tank, when a solid deposit is made, you want it to stay afloat, not be stuck to the floor where it will build up into a pyramid of poo. When emptying, do the black first and then use the gray to flush any of the residue from the black out of the hose.