That does sound like a confusing mess. I'm no expert, but it appears to me you've done all kinds of right things.
You say they swarmed a month ago? Was wondering if the queen you now have is a purchased queen or one the bees raised. Could she be just young? And that would account for the multipule eggs in cells? The hive is on the edge of honey bound, but I know you're doing alot to fight against that.
I'm thinking.... things will eventually get better. BUT, in the meantime you want to do what you can to get any capped honey out. The goal I'm thinking of is to give room for laying. I believe I'd extract honey, and give what frames and foundation on the lowest box. I think the bees for now are going to put the honey in the first cells they can (lowest area). But maybe if they do, the queen can find somewhere to lay higher up.
I know they'll get it all straight at some point. I had a hive get honey bound, but after awhile they fixed it.
It seems to me that you've done (and/or thought of) all these things yourself. So I may not be giving you much new to work on.