You have not told us one very important thing: are they neutered? If not...why not?
Spraying is a normal behaviour for an uneutered cat and he WILL do it! But I'm assuming you know that and he is neutered and if the vet gave you all clear, it's not health related.
Have you talked to the previous owners? Did he have this problem at their house?
IMO I think it's stress related and he might not yet be comfortable enough. Also, try to put a litter in your bedroom for a while, see if he uses that (the litter might be just too far away, and the bed's nicer and closer).
It might also be teritorial if you say the neighbours cats are nosing around. Can he see them? Have you noticed how he reacts?
In regards to going to the litter after eating/drinking, I don't think there's an all around valid answer, it really depends on the cat. Mine will sometimes go just after eating, or after 5 hours. Sometimes I clean his litter and he walks by and is all "Hm, oh yeah, I could use that" and jumps in and pees. With kittens it's a bit more predictable because they can't hold it that long.
They should have water at all times, unrestricted.
You can try again to confine them to a room, but make sure they don't feel like they are locked away.
Good luck!