Welcome back Deli!
Omni, Kioka and Liyure are still the most valuable pets as they are only available to get through a special promotional event. But pets that are sold through the Kir Quest are also valuable as they required a certain amount of points through the Kir Quest.
Yes, but names can be put back into the system by either a name tag from the Credit Shop or from another player or if a pet is left dead for 15 days in someone graveyard, they will lose their name.
For the Item Auction section yes. For pets, you can use the Creatu Search system under the Information drop down menu and under location make sure to have Pet Auctions selected.
Same thing, searching for rancher shops. Just use the Creatu Search and have Rancher Shops selected instead.
Hopefully I answer all your question completely. :)