How can I empty a bucket of stinky water under my sink?
You should siphon the water out using a tube or flexible pipe.  I have included an article containing instructions on how to do this here. You will have to adapt it to a much smaller scale, but the principle is the same.  If you just do a search for siphoning, maybe you will find a better/more applicable example. I just chose this article bc it wasn't about stealing gasoline. Let's not promote that slimy practice, shall we?

I had to do this once when the grease trap got clogged at work backed up into the sink. It was 11 pm and I had to get the "water" out of the sink so it wouldn't sit there all night and stink up the whole place. There was no WAY I was sticking my hand in there to scoop it out. Siphoning works. Give it a try.