Tesla service has asked me before if they could take my car on an errand. I was happy to say yes, and it's obvious that others here are too - though clearly not everybody. If they'd consistently ask, they could separate the two groups and everybody would be happy.
It's not just that I'm "OK" with it; I would rather have them to take my car on an errand (assuming they have to drive the car anyway). As long as they have to drive my car, I'd prefer it to be a useful trip that eliminates another trip they'd have to take later.
Asking first would solve most problems - though not all. When the car gets dropped off in the morning, they probably don't know which car will be ready at the time that a specific errand will be run, and some people may allow certain errands but not others (say, dropping off a co-worker to pick up another car for service might be OK, but they don't want any food in the car). Covering all possibilities in advance may be difficult for both sides, just as calling at the time of the errand may be inconvenient.
As for dings, yeah, you should check that any time anybody has your car, no matter who or where they take it. Tesla service did a very minor scrape on my car once, but it wasn't on an errand - it was while moving the car in to position for service. They told me before I picked up the car (which was nice as I likely would not have noticed it until the next time I washed the car; then I wouldn't have known who did it), and ordered a new part to replace the scratched piece. I don't think where they take the car significantly changes the fact that this can happen or how they should handle it when it does happen.