Shipping for inoperable (INOP) vehicles adds about 10-15% usually. It's not insane, but it is more, and does add some lead time depending on location. Fortunately, not needed for the return trip.
As for the screens... it's actually pretty random and doesn't seem to have much to do with heat. In fact, I just replaced my own S instrument cluster over the summer and it's been below 50 daily, and the replacement one has just now started to bubble again.
(I also don't have the bezel installed, so I can see the edge where it's just beginning...)
Shipping generally isn't much of an issue as long as you're not the person who just absolutely demands it be picked up that second or otherwise has unreasonable expectations for things. I've actually turned away a handful of people over the years over this. "Hello, we're shooting to get your car picked up before the end of the week. As soon as a carrier has space and route availability, we'll get you more details and put you in touch directly with the carrier to coordinate a time that works for you." "(at 4PM) I need it picked up today, before 9PM." "That's actually not going to be possible." "REEEEEEEEEEEEE".
Truckers are fickle. They're rarely on time, and rarely stick to a schedule at all. So it takes some effort on our end to round up a carrier for each vehicle. But once we lock in a carrier for someone, they'll definitely show up and your car will make it to where it needs to go. Most of them will also work around reasonable pickup/delivery time windows and such with customers as well.
INOP vehicles almost always take a little longer to schedule, especially for further hauls, since they're more hassle for the carrier. Generally they'll have to make it their first or second pickup, depending on the truck type, and usually one of their last dropoffs since they won't be able to shuffle it around like the can with operable cars on a multi-car carrier. So this limits the available spaces for pickup and adds some lead time. But again, once we've got the carrier on board, they'll show up, winch it up, and get it on its way. We occasionally will walk a driver through procedures specific to Teslas, but we usually ask the customer to put the car in tow mode and chock the wheels as soon as possible after a failure since this will make it simpler. Our crew unloads the cars, and we're pretty well versed at this point.
Then it's operable for the return trip, so generally no issues there.