Hi Marigold-Mushroom, I too play at Magistream and Dragon Cave, but I play at these sites too, if you want to check them out.
http://taleofostlea.net/ - more relaxed game, there is a nursery where other people can click your eggs/hatchling to grow them and you can click others eggs to make the site currency,
new pets are released irregularly, events are mostly about releasing event pets,
pets are breedable, most of the event pets are breedable only during an event
https://eggcave.com/ - 4 new pets every month exclusive to that month, 2 found in Cave and 2 for Cash, and some pets are found only during events
you can have 1 main account and another 3 side accounts - but you can participate in events or explore the Cave (where you can find pets) only from your main account,
each account has free 300 cove spaces for your pets - you can expand you cove
your pets can die if not fed regularly until they reach level 2000 than they become immortal or you can buy the Leaf of Vaka in the Cash Shop it protects you pets from dying on one account - side account are good for storing immortal pets to free space on your main account
http://www.rescreatu.com/ - different locations you can search for different pets,
if you don't feed your pets they will die but you can keep them in a showroom where they are safe,
fun events that last a long time so you can participate properly (for example the Valentine event ended just yesterday),
you can sell pets/items in your shops,
this site is more about items that pets - there is about 2 new pets released per year, but there are pets that are found only during events that return every year
http://www.aywas.com/ - different locations you can search for different pets, events, you can sell pets/items in your shops, if you log in every day you get the site currency - the amount is increasing every day (I'm at sum over 15k)
https://gothicat-world.com/ - it's French but you can switch to English,
new pets every month like at ChickenSmoothie, there are events with event pets,
you can trade pets with other people - the trading has a feature that tells you if a trade is fair