I believe this has come up before but here it is again.

I just joined a battle and I want an opponent. Could the Who's Online list show that the player is ready for any opponent or is waiting for a specific player?

That way one doesn't have to beg for a partner or put up with an interloper when waiting for someone specific.

Could there be a check box(s) at the start of the battle to indicate this?

When you go to your battle, and click on 'join game', if gives you the option to select a box. One is titled "open" and will allow anyone to join, whereas the other is titled "send invitation" and only the person you send the invitation to may enter.

The "who's online" list can have 2 different icons to use beside the "auto battle 5" (or whatever battle your in) to indicate to others if it's "open" or not.