There are multiple animals in my stable/kennel that, when I click on them, are "invalid" but they're still taking up spots in my stable/kennel and since board is paid I can't kick them out or anything.

http://prntscr.com/5auwd0 -- if I click on the first animal there, dog #53512 (http://horsephenomena.com/dogs.php?id=53512) the page says invalid animal.

If the animal isn't on the game, why is it still being boarded?

It's not an invalid animal in the database, the main reason why this happens is that the account they were originally boarded on no longer exists or the account which the animal is existing on no longer exists. In this case, the owner no longer exists. I've deleted it.