Don't know if anyone has mentioned this, but I think what may also be a nice option is when a pet is already in a trade, that when other users are selecting a pet, for it to show them that the pet is currently in a trade.
I know it does this on the user's own side, but in the many trades I've done now, don't think this applies the other way around yet. May be more work to code for this?
Or, maybe, if someone is currently actively selecting pets, to have a lock up feature for a pet that's been added so other users who are also simultaneously trying to select pets aren't able to do so for that pet. (Kind of like when you're in the process of buying movie tickets and the system "reserves" the tickets for you so that it doesn't run out while your checking out).
Now, I feel like this path would be majorly complicated to code and implemented, but if successful I think could help make trading much better and smoother for both parties. Since, as the person initiating the trade request, I would be able to know which pets are already in a trade and then I can select something else, instead of later finding out it's been auto cancelled and I'd have to send another trade (and all the while again wondering if someone has already picked it. Particularly an issue for players giving away free pets, since I bet there's a large volume of trade requests going down simultaneously).
Edit: Also found other related suggestions
Trade counter for petStill be able to accept trade