They were stating that for a specific reason. "they could ask for proof of age ...to be in lap during these times"...
You would be amazed what people do, even when their kids have seats, on planes.
I assume you mean those weird horse shaped things? Those look like such a pain. Sending the stroller through the entire "innards" of the airport increases the chances of a mangled stroller. Forget the horsey roller"bag", use the stroller.
Well, being nervous (now or on the day) isn't going to help one little bit. You need to figure out what you will do. I don't know the circumstances that led them to have you hand the baby to a stranger. Perhaps they thought the stranger was with you. I don't know. But what YOU can do is stay CALM. Don't be nervous, don't act nervous. Be Zen. Even if they are moving quickly, you slow them down. If they are talking about moving you away from the child, state firmly and clearly that it's just you, and that you have to stay together. Just be calm.
And make a list of all those *library* books so you're sure to pack them ALL up when getting off the plane each time and when coming home. No need to have a fine!