For rotisserie mounting, the normal place for the front is on the outer edge of the frame rails where the bumper support arms normally attach. For the rear, there are 2 normal places - either the leaf spring mounting points or the mounting points for the rear bumper brackets. It would work for a body cart as well, but if you are wanting to keep those areas open for painting, then mounting to the transmission crossmember and the leaf spring mounts would work fine. Those points are strong enough to take the weight.
As far as putting it on a trailer, I would ensure that the body cart wheels can't roll by screwing some wood to the trailer deck around the wheels tos erve as chocks that won't go anywhere. Then I would strap everything down really well using heavy duty ratchet straps (not the little bitty ones, but ones that are strong enough to hold a car). I would probably add some temporary wood cribbing under the frame rails to support the body so that it is more firmly fixed to the trailer than just the 4 mounting points to the cart. The ride on a trailer can be pretty rough, depending on the condition of the road.
No worries with moving it on a trailer as long as it is properly tied down.
There was a thread here not too long ago dicussing transportation of a body on a trailer while on a rotisserie - a body cart would be basically the same thing without the ability to rotate the body.