There is a better way. There's a reason the top of my transmission is still off.
As for draining it I did it with two oil catch pans. Stuck the plug back in between pan swaps. Dump one into the 5 gal bucket while the other is filling. It's a little tricky getting the plug back in but I've had plenty of practice (used to work at a quick lube) and the gloves make it easier.
The only problem I have with the oil extractor is it doesn't clean out as much of the gunk sitting on the bottom of the case as draining through the plug does. To drain differentials that didn't have a drain plug we would hook up a line to our diaphragm pump and do it that way. We experimented once on the differential of my coworker's truck. Drained it with the pump and then pulled the drain plug to see what was it didn't pull out. We got a quarter cup of sludgey gear oil that had plenty of suspended metal particles in it. I'd recommend doing the last few quarts with the drain plug if you use an extractor.