Yes, the condition of flanking, per my interpretation, is a distraction on the creature from two or more hostile creatures roughly opposing the target. However, I allow advantage on grappled targets, and the grappled targets get disadvantage but flanking will cancel that out.
Also yes, the bulette did lose his disadvantage because of flanking skeletons. He can't have advantage because of Alice's grapple.
Necrotic damage cannot awaken undead, but in this case it did. You did not supply the energy, mithril ore apparently does that, but your choice of flavor of damage resonated with them.
The bulette will attack whatever he wants.
Cat can also grapple it, but there is no additional effect other than the bulette will need to get both of them off to move normally; however, realism is sometimes used. Imagine the notion that a <100 girl can "grapple" a >2000 beast. Don't rely on vanilla hard and fast rules with me.