For arena battles, would it be possible to add an option of fighting a random opponent (randomly chosen from among all the currently available ones)?

Something like 'Accept Random' button, next to 'Accept Challenge'?

I'd prefer to use that myself instead of manually choosing the opponents, since it'd make the battles more arena-like: you'd never know who you end up battling. And it'd feel more fair than beating the same weak pet ten times in a row.

Would also save some time, with the downside that you're more likely to lose when you're not hand-picking the battles. 

I like this idea. I was thinking about the new arena system earlier and commented in the latest news post. The new function protects the average weak (read: non-GP) pet by sheer numbers, but that may not be enough. In the past I've just had to train or manually battle my non-GP pets if I wanted to get anything but losses. Depending on how this new system plays out - which kind of depends on how many people purchase premium to begin with - a randomized system like you suggest might be needed.

Building on that idea: what about the arena functioning like the Pound?