My Puma was sent out on a quest about three days ago I believe, and I am still unable to use her. It says she is 'exploring the island'; I'm new so I'm sorry if this isn't a bug but shouldn't I be able to use her again by now? All other quests have ended within 24 hours for me. 

For the tutorial, I was in the middle of it and it asked me to 'research all genes'. Before I had even done that, the tutorial disappeared? I did end up doing the 'research all', but it was suggested to me that I post this here in case that wasn't the actual end of the tutorial. 

As for the Puma, this isn't a bug. The exploring status means that your pet is still on an island. While the Puma is active (go to its Profile page to make it active), click on Explore. Usually this would take you to the page where you click the Portal and select which island to go to. With this pet though, the link will take you directly to the island that this pet is stuck on. You can then leave the island via the portal link on the right side, which should make your pet available again. 

Also just a minor note, quests take 4 hours. You can see how long a pet will still be busy with it if you are on its profile page. In the little box on the right where it says "Effects", it will tell you how much longer the pet will be on its quest. :)

Timain has already provided a full answer for the exploring issue, there are no bugs here. Thank you for help, Timain!

As for the tutorial, "Research all" task is indeed the last tutorial tier.