It sounds like it was, as you say a mostly good outcome. Was the dog not checked all the way to ORD? That part of your post is a little confusing. It sounds like the dog needed to clear customs and that wasn't going to happen without your sister. I don't know but is a pet being checked like a bag where you have to claim you bag/pet at first entry point then re-check? Your sister claimed her 3 bags and rechecked but not the dog? I'm a little confused by this part of the situation. It sounds like UA bent over backwards to make things right, including rebooking your mother and giving her an e-cert even though on a totally separate reservation.