I skimmed through your posts to see if there was any sign of your H being in AA...I know he's in rehab, but that's just for 28 days. What's to happen afterwards? If he is serious about his recovery, he should be in a support program for a long while, and you'd definitely see the signs of this. Also, were I in your shoes, I'd demand to see 6 months of sobriety before making any changes to visitations or putting any further trust in him.
Sadly, there will probably be quite a bit of "booze policing" on your part to make sure that he hasn't fallen off the wagon or isn't drinking while caring for his children.
I also wanted to speak to the suicide threat...if it ever happens again, call the police, and send them to his house. If he is serious, then you are not equipped to handle that sort of emergency. If he isn't serious and is using that threat as a means to get attention, then he'll think twice about doing it again.