I tried to use one of my pets on a quest and it says that it was busy researching, so I went to get rid of my squad, but then it said that my pets were on my battlefield.  So I went to check if my pets were on the battlefield, and they weren't.  I haven't been able to use these pets for about a week now, and I'm thinking something is wrong.  I'm sorry if this isn't a real bug, but I want to figure out if something is wrong.

As I see, your squad is on the battlefield now and it is researching the way to the mirror world at this moment. All you need to do to free your pets is leave the battlefield and stop reserching with this squad by clicking "Leave" button on the tactic map. Researching is an activity just like any other: it locks the pet until it's finished or cancelled.

I just went to the Homeworld Battlefield and there are no options for taking my pets off.  I tried reloading the page, but it still won't show up...  Am I missing something?

You can't research the way to homeworld, it's available by default without researching. You need to go to mirror, your squad is there.