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.

[Ashley] If anything,  I'm putting the heads back on the trolls. And you're totally wasting our fodder by taking out the skeletons. Are you nuts?

Alice grapples the beast and begins her attack. Alice uses Flurry of blows for 1 Ki point for a bonus action and attack with an unarmed strike as well as Hand of Harm (2nd Ki point).
Alice adeptly lands every blow, she couldn't have had a better attack, but the creature resists a fair bit of her bludgeoning damage for a total of 16 damage.

Damn, so it resists bludgeoning damage as well!

With a near deafening bell that kicks up dust and splashes green goo, Yulya's spell rings true on the beast dealing a total of 20 damage on it. However, the necrotic damage seems to resonate with the skeletons in the chamber, eight small skeletons awaken and begin to shift. Wielding whatever they can manage to grasp, they join the battle.

Hey hey I told you this would happen! At least we took put the big guys I hope!
If she can, Alice attempts to shove the bulette prone. If impossible she attacks it with her spear 1-handed. Alice uses her bonus unarmed strike to hit the skeleton N of her with bludgeoning damage. With her Crusher feat, she moves it 5ft W towards the hole. Maybe the skull will roll away and down into the abyss? If possible, she repositions herself on the bulette to be within Cat's Sentinel range, but maintaining a grapple and staying away from the beast's jaws has priority.

Yulya's spell didn't animate them, but it did spark them to unlife given they were already primed. Whether they animated or not was based otherwise on chance, but undead love necrotic damage.

You got 11+1 yes

Oh, quite a lot happened! Wait, Yulya's spell animated the skeletons? Or the green goo?
So Cat was bitten by the bulette. Didnt see that at first because it was not red. 1 DMG from skeleton and 11 from the not-gopher, correct?