When it comes to Kenta's Advice and the quest at Henesys Pet Park that allows you to have 3 pets at once, the Retriever donor pet is *not* counting as a pet. It was only when I equipped my pet pig that it allowed me to get the quest from Pet Instructor Bartos to do the jump quest and exchange a pet snack. The quest, Kenta's Advice, only exists when I have my non-donor pets equipped. It is not registering the Retrievers as pets.
I believe this applies to lil aran also - I got my closeness level to 16 with just my lil aran out and no quest was available.
Quests related to the new pets were forgotten in the recent patch. We will look into getting this fixed as soon as we can, but I am pretty sure it will require new files.