This sounds like a really messed up situation. Sorry to hear you guys are going through this.
I know the answer to the first 2, not the third.
When I was 17, my parents and I moved to Hilliard and my school district (SWCS) found out about it and tried to kick me out of Westland (lol) for my senior year due to being in the Hilliard Darby area. Long story short, my parents gave custody to my at-the-time girlfriend's parents (for approximately 6 months until I turned 18). This allowed me to graduate with all the kids I had grown up with.
So yes, legal custody can be awarded willingly. It was not difficult, but we did have to appear before a judge, etc.
As far as number 2 goes, yes, a legal guardian can be put on your insurance. As long as you have the documentation to show they are a dependent, you are fine. Since this is a qualifying life event, you will not have to wait until open enrollment either.