I always recommend (if it is an option) to get your non-running car working before shipping it. Obviously this will save you money but it also saves us both time. It’s much easier to find a carrier and if the car is located in an unpopular area they can be near impossible to find. And it doesn’t need to be working perfect either, it just has to be able to limp onto the trailer under it’s own power.
Washing your car is usually pointless, keep in mind that your car does need to be clean enough for the driver to be able to do an inspection though. But cleaning it in the hopes that it arrives at it’s destination clean (especially on an open trailer) is time wasted. Obviously enclosed trailers are much better than this but the car will arrive at least a little dusty either way.
Topping off fluids beforehand is a good idea. But make sure to double check them once the car arrives. Your car may be at an angle for an extended period and sometimes this can cause some fluid to leak out. It’s better to be safe than sorry when dealing with vital fluids inside your car.
Your tires will lose pressure during transport. Make sure to refill them to the proper psi once the car arrives. Both ways of securing your car to the trailer (wheel straps and chains) put more than normal force on the tires for an extended period of time. Your tires won’t get damaged but air will seep out, especially on longer trips.
These are the most common things people ask me about. A customer once asked about putting a giant bow on their car before we picked it up (it was a gift). We had to tell him no in case it flew off while in transit but if there was a bow that was made specifically for cars on trailers we’d do it on every car.