Question on how this will work (Divorced / anticipating new teen drivers)

I called my insurance company on this last year when they got their permits, and because in my state, they do not need to be / are not specified on my insurance policy, they were unable to even give me an idea of what to expect – now that they will soon be licensed, trying to figure out how insurance works for children with divorced parents.

I have two kids (twins) that are eligible for their licenses 3 months from now. My ex and I share 50/50 custody (though I think for the 'technical' legal purposes I am the custodial parent (I think it mostly only comes into play as to which schools they attend). Right now, I have one car (for me, single), he has two cars (his and his wife's). I am planning to buy a new car (for me) probably around their birthday, and plan to keep my old one for their use. (Not sure how it works in his house, but obviously, my car is the only one available to drive in my house currently. After I buy myself a new car, they would be listed as allowed drivers on the new car.)

Once they get their licenses:

Before I buy the new car:
Who insures the kids? (assuming that they have "occasional" use of any of the cars between our houses, but only during "our" time (e.g. they have access to my car when they're with me, his two cars when they're with him)
What should I expect rate increase wise? (they are girls, and fwiw, get good grades. I'm currently paying ~400/6 months, for a 15 year old minivan. I think I still have collision on it… should probably drop that)

After I buy my new car (my old minivan will still be in my name, but my kids will be the primary drivers):
Is there any good way to separate what costs are related to 'insuring the additional car + having the kids on my policy'? (or since we're still 50/50-ish, is he going to have to also cover them on his policy as well, and then it's just the added cost of the 2nd car?)

What happens if – after I have this set up – if dad does something similar (upgrades one car, and keeps the older one for "their" use)? (this seems silly since they do not need two cars, but…