Hi Puddle! Things seem to be moving pretty fast for you but carry on regardless - he is the confused one. Great that he accepted your boundaries about the dates/phone. What a goal you scored.
I think the basement idea stinks. Would you be prepared to live like that?
As far as taking the kids out with mutual friends, I think that's grossly unfair on you and the kids. It'll only cause more confusion all round and may distress the kids, depending on H behaviour.
On the plus side, he has accepted some boundaries, maybe you could work out some more regarding the kids?