I used Montway to move a car from CA to TX. Like others have said, unless you're using a dedicated auto transport company like Reliable or one of those guys, it will be brokered. The brokers will give options for enclosed transport, though. The dedicated, enclosed guys are 2-3x the cost of brokered transport, which could be worth it if you have something high dollar and need to pick up at X time and have it delivered at Y time.
Montway and others will give a "window" for pick up and delivery. They can give a dedicated pick up time, but it costs a little extra. I opted for the dedicated pick-up time, and in total spent $1,125. That was about 1.5 years ago. I'm not sure if they can guarantee a delivery time. It largely depends what all is on the trailer and where you fit on the map. But, they'll probably get it to CA within 3-4 days.
Also, you don't pay until the vehicle is received. A deposit is reasonable (I think I may have put down $250, and that may have been for the dedicated pick up), but a good company will have you pay upon final delivery, not up front.