Somehow I missed your reply - sorry about that!! ^^;
Anyhow - nope, the pet that you got was a non-combat pet, so it couldn't get knocked out even if you tried. Even though, I changed that in the last two weeks: Now your pet actually helps you in battles, and hence can be attacked and get KO'ed - but you also get a skill to revive it in case that happens. ;3
Naw, she should drop her hat. I've seen it before - it's just much rarer (10% chance) than her other drops (45%). Then again, it looks pretty good on most characters, and has decent stats, so it's worth it. >3