I tell the rescuee (the customer) that if I show up they have to pay regardless of whether the cat jumps, downclimbs or I fetch it. The truth is, it's the owner being rescued from their worry about the cat, that's why I prefer to call it fetching.
I agree about non or partially domestic cats, they bite and it's no small deal when they bite hard. I know a woman who lost 2 fingers from infection after her own cat bit her. It's also iffy about who gets the cat, so far I've done two like that and the people who called took the cats to animal control.
I like the idea of charging mileage, that's where I lose money on cats... Raleigh is 45 mins to an hour away and I've been turning those down lately.
I bag'em in a pillowcase, I can't get a good grip on the scruff with gloves.