I don't think anything can be done about the problem in the OP because it sounds like a one-time fluke :/ Sometimes things happen like little site hiccups like that. If it ever affects your items or anything, you can make a ticket and have the mistake corrected - We're happy to reset quests, reimburse lost items/sP, things like that if it was the fault of a site glitch.

For the other problem being reported, it sounds like it is more common but still kind of hard to trigger/reproduce. I wonder though if this one is related to how aggressively cached the battle start page is. For instance, if your pet gets knocked out and then you heal it, it might not show up still because that page still has it cached at 0hp. So maybe it's doing the same sort of thing there, is what I am speculating... I'll see if I can fix that cache, it might not be related but maybe it will help? We'll have to see D: