It might be a mobile problem, but I'm not seeing any battle opponents in the battle tab. I could see a dozen or so yesterday on my laptop, but maybe the script won't load on a mobile device. Anyone else have this problem or know what to do?

There just aren't any pets in the Battle tab, haha. The way the battle tab works as far as I know, is that people can put in pets for battle. Basically, other Keepers put their pets up for battle, and this allows you to fight them {i believe} until their energy runs out or the pet is taken out of fighting. Once that happens, the pet disappears off the battle tab. 

Sometimes it does run blank because no pets have been put in, or all pets have been defeated.

{Though there are a few that have been put in now, so. xD}

Makes sense :) During the tutorial they had quite a few "Nameless" owned by BeastKeeper, so I figured those would always be there. 

Sometimes our system user lets out his pets to fight some battles on the Arena but usually not more than twice a day; it would be not nice to make pets fight the whole day without rest. They go to rest just like any normal pets to regenerate their energy.