I'd be concerned with my children visiting their father in that environment too. It's great that everyone there is working on their addictions but it's not a place for kids particularly when there are no background checks. Considering that background checks are required for virtually everything these days if kids are involved, that sounds concerning to me as a mom.
As for your question of can someone be addicted to AA. I think that anyone with an addiction probably has a predisposition to move from one addiction to the next and there may well be people who get addicted to a program in an unhealthy way as opposed to working a program and pushing themself to use it healthily... Not sure if that's the case with your H, but the fact that he's concerning himself with what he thinks you should be doing with al anon, isn't willing to make a decision or communicate at least with you about the direction of your r/s, insists on keeping his visits with the kids in a place that you have good reason to be concerned about... those seem like worrysome signs to me... But that's just my 2 cents...
Sorry to hear of your situation-- I think that during the one time my H made a seemingly sincere effort with recovery I had hoped that he'd do an about face with lots of troubling behaviors. At the time it seemed that he replaced not drinking with a bunch of other addictive, haughty behaviors and I'm not sure he ever really got on board with the recovery thing... As for your H's sponsor and you asking if his sponsor approves of or would say anything to your H about some of what you describe... what I noticed with my H is that when he wasn't genuinely invested in recovery it really didn't matter one bit if he had a sponsor. He's had sponsors and b.s'd to them to make them think what he wanted... Alcoholics are con artists (my T's description and he has worked with A's and their families for 30 yrs). They will and can con anyone and until they decide they want to live differently it really won't matter whether they have a sponsor or not-- the sponsor is there to support the willingness that has to exist in the person who is recovering.
Just based on reading what you wrote it sounds like there's a lot of addict thinking still going on with your H and addict behaviors. It sounds from reading what you wrote that your H is using Oxford house as a hideout to not have to deal with any of the parts of life that he doens't choose to and maybe that is a part of recovery but it sure sounds frustrating to be on the spouse side of...
If I were you I'd talk to someone legally about whether you can require that your H see the kids somewhere other than the rehab house. That makes me very nervous too.