some questions about shipping a car

I've looked at the websites for many of the car shippers recommended here, and I noticed that some seem to be brokers and some (not sure what the correct term is) companies that actually do the shipping themselves.

If that's correct, is there a major difference in how it works / a reason to pick one arrangement over the other? If I had to guess I'd say dealing directly with a shipper would mean fewer complications/fewer chances to blame a problem on some third party, but I have no idea if that's true IRL.

Second question: any sense of how often it happens that a car shipment is seriously delayed?

Third and final: the car has to be empty, right? there's no shipping co. that will let you pack stuff in it, because this is actually a law and not individual company policy? (I'd love to be wrong about this but I'm not holding my breath; either way, confirmation would be helpful.)

Many thanks!