I know it's more realistic, but it is extremely irritating to try breeding with another user's pet, only to get a mission error. I feel that this is a step backwards, especially since they're invisibly unavailable, it's like guessing in the dark trying to find the right minute between them finishing one mission and starting another.

I suggest that either pets on a mission by other people are still allowed to be bred with, or else take them off the list of available breeders for the duration of their missions. Because finding the perfect mate, only to have no way of knowing when they'll actually be available is not fun.