According to my records, Wash never took Villagers' Trust I, which is the prerequisite perk -- so he can take that, but he'll have to wait until after the next plot round to take II.

They should definitely aim to be more discreet than occupying an empty house (which won't stay empty for long). Dead drops are a much safer bet, but if they're going to be meeting in person, it needs to be somewhere more discreet, and if they're meeting in person very often, it will affect Natasha's cover roll, since it puts her at higher risk for discovery.