No and no. I was told, any animal taken outside of season has to be left at/near catch site. I have done a few jobs on private land where the DNR picked up beavers I had trapped for state training/demos.
You are correct with the no and no. Every time I trapped beaver out of season, I was reminded that I couldn't take any part of the animal whether it be fur, castors, tail, etc.
I had one job where beaver had dammed up a channel flooding the farmer's field. I trapped 5 beaver and left them lay. Later, I got a call from the farmer mad that I left the beaver on his land to rot. I told him by law I couldn't take them off the site. If I got caught with them in my truck, I'd get a ticket. He said that was the stupidest thing he'd ever heard of.
I called the CO who told me I couldn't keep them and asked he what I was supposed to do with them? She said I should have buried them on site. It's really pretty stupid because the fur is pretty much worthless in the warmer months.