No you don’t want to get back with him just get back with him just cos it’s the easy thing to do. You should see this as an opportunity to rebuild a new R that will be more fulfilling for both of you.
Interesting that he’s stepped up to being a father now that he has the kids on his own. Perhaps you did too much when you were together? This is a form of you pulling back and he’s stepped up do you think?
As for the ‘date’ conversation it sounds like he’s trying to stir it up a bit. I think sometimes that if the H feels that things are getting cosy again he pulls back, remember a reasonable conversation may make him feel guilty, it’s like he’s cheating on OW. His loyalties are split.
Why would he not want anyone else to know that you were staying at the house? Split loyalies again? You’ve asked to spend time with him, see if he picks up on the invitation and takes the initiative.