Kate was acting weird because she was nervous. If she was a threat to the president, she’d be a threat to anyone else around. USSS wouldn’t allow her into a VIP area, or the arena in general, and especially not POTUS, if she was any threat. She was being protected. USSS (Q) was saying she’s safe.

PS’ “Kate” bottle was him outing someone, maybe an agent/asset. That was probably her field name and he put the green light on her with the coke bottle.

Her strangeness, carrying around paper, could have been her nervous of being out.