I've had this happen to me a couple times; I am testing breed for a couple of my pairs, and then suddenly the preview for the child disappears. Refreshing the "test breed" option does nothing to solve this, but it is still possible to breed pairs while this happens.
The only way I've been able to solve this is by leaving the page and returning to it, but it's happened enough times now that I don't feel like this is an accident.
It's not terribly major or anything - it's just a minor inconvenience. :P



Upon changing species and pressing "test breed":

Is there anything specific that causes this to happen? Like if you do specific pets, or take a specific action.

Will be extremely hard to diagnose without further information to try and replicate it on my end.

There is nothing specific I have seen about it - it randomly happens sometimes when I am breeding pets, after I have pressed the "test breed" button many times. I wish I could give more information, but I really haven't seen anything else similar in the situations where I encountered this :(
It has not been any certain pets or anything, either.

Unfortunately, I am not able to reproduce this issue.

My best guess is maybe every once in awhile your connection to the site gets interrupted while trying to load the results and then the script just enters a permanent state of limbo until the page is reloaded.

If that's the case, there isn't much that I can do in the Breeding Ground's current state. But I can say that the Breeding Ground is near the top of the list for being completely overhauled. I would imagine any issue like this would be resolved at that point with a complete redesign.