When I purchased our 2012 S P85, Tesla shipped it from Fremont directly to my front door in Houston (the nearest Service Center was in Denver).
Even though it came in an closed carrier, Tesla still had a detailer come to do a thorough cleaning of the vehicle after delivery. Though with an open carrier, it would likely have been a little more dirtier on the outside.
I don't have a recommendation on the carrier, though when I've shipped vehicles from Houston to Denver or San Francisco, I went onto one of the vehicle shipping websites, submitted my shipping request, and then waited for a carrier to indicate they'd be able to pick up the vehicle. There are a lot of independent transporters - if you have scheduling constraints (when to pickup or to deliver), you may have fewer choices on price and type of carrier. If you're flexible on timing, you should have more choices.
As for the amount of battery charge - the vehicle will be driven only to get on and off the transport. It will be exposed to outside temperatures - which will incur some battery use in order to keep the battery pack in the operating temperature range. Though this shouldn't be any different than parking the vehicle for an extended period outside. If the vehicle starts at 90% charge you should be OK, as long as you've enabled the energy saving features (depending upon MCU1 or MCU2) - disable sentry, disable "always connected", ... This is how Tesla ships new vehicles - across the US - and on ships.
