I always bring mine in, in their crates. They have things in there that they're familiar with...a blankie, a chewie and a toy. The crate is clearly marked with my and their names, phone number and address, and an emergency contact #. Because the Vet techs really like my dogs and want to play with them, they're usually taken out of the crate so there's some time for interaction before they're carried to the cage room. When they've had surgery, (Tango's baby teeth extractions, Jazz's spay etc.) I always put them in their crates for the ride home if they're not brought to me at the front desk for pickup already crated. If I had someone else driving, I'd just hold them, but I don't. So they're crated so they're jostled as little as possible during the drive, both to cause them as little pain as possible, and also to (hopefully) prevent any vomiting. I don't know about you, but I simply can't operate the car pedals with a dog on my lap WITHOUT jostling them somewhat.
My vet puts their toy and blankie in the cage with them while they're waiting for surgery, and when they're recovering. They believe it comforts them, and I do too. Those things have all the right smells on them, the comforting smells, of them, their sibling, me, home, etc.
Personally, I don't favor putting tea tree oil on the incision, putting anything on the incision, unless I'm directed to by the Vet, When there was a slight infection after Jazz's spay, I took her back for antibiotic shot and a course of oral antibiotics, and was told, and shown how to put just the tiniest bit of antibiotic ointment directly on the infected part, but nowhere else. I use bitter orange to inhibit licking, supplied by the vet, and I smear that all around the incision but never on it. When it dries, it's chalky but stays on a good while. One lick of that and Jazz didn't touch her incision again, at ALL. It's nasty stuff! I much prefer using the bitter orange to having to use an e-collar. It's been impossible to find one to fit my dogs, the Vet's had to sort of retro-fit the smallest one they had to sort of work. But the bitter orange did the trick, so I didn't have to resort to the e-collar anyway.