That's a nice idea, but "unwanted pets" are called "duplicates", and normally go into the Trade Clan. In this way, others know that they're up for trade, and can come to your trades to propose trading pets you don't want, for pets you do want. They'll know which pets you do want if you fill out your wish list.
We're allowed thirty free pets a month - six of each of the five breeds. This doesn't include any site events, such as the advent calendar this month, or the easter egg hunt, the halloween potion event, or other such things. That's why the tab for acquiring your creatures (always given out as eggs; they mature throughout the month) is called "adopt" - that is, essentially, the adoption center.
I hope this helps you understand a bit better