Hi there,
Pets in groups are organized by ID (which is assigned in the order they were adopted from the site originally -- each new pet has one higher ID number). Even when you put pets in different groups, all of the open groups will appear mixed together on your partner's wishlist. They will be organized by ID here as well, so in 99% of cases, multiple copies of the same pet should be next to, or at least near, eachother. Your trade partner will be able to see the name of the group the pet is actually in though, in case that has important information or they just want to see where it came from.
There are some exceptions though. Re released pets have higher IDs that their true date counterparts, and will sit far higher in the list because of that. Similarly, there are some pets that may have been recreated by admins for one reason or another, which will also have a higher ID.
Locked groups do not display the same way. Only you can pick from a locked group, and they display at the back of your partner's wishlist. Meaning, you see all the unlocked pets first (organized by ID) and then all of your locked pets (organized by ID again). This way you still have the option to offer them if you really want to, but other cannot and they are out of your way if you do not want to trade them.
When you open your trade center, you can edit your trade rules by clicking the "trade settings" button. Here's a direct link:
trades/tradingsettings.phpThere is no way to preview your trade center exactly how others would see it, but you are always free to ask for help! I'm sure someone would always be able to screen shot what they see for you. c: