The 2 spigots with drain in between is a common approach for washer/dryer hookup. You are correct that the washer drain hose just goes into the drain hole. I should have a trap built in it some where so a trap shouldn't be necessary in the sink. In every case where I've seen this hookup, it's been relatively high (about 36"). So it could be difficult to raise the sink up high enough.

I don't think you need a specialized grey water pump; any small, submersible sump pump should do (smaller the better). Ideally with a garden hose adapter. Maybe even a cheapie fountain pump like this: (just don't expect it to last long).

In all likelihood, the spigots are 3/4" threaded and the faucet is 1/2" threaded. Washer hoses are typically 3/4" on both ends, so an adapter for one end will be necessary.