In response to the dethrone ability, Bob casts a Dismember targeting the Bear, and paying 4 life. He is now at 13, less life than Alice. Does the dethrone ability still give the Cutthroat a counter?
Once an ability has been triggered, that's it: it's on the stack and you don't have to worry about the triggering condition anymore.
You can see this in the Gatherer rulings for Grenzo's Cutthroat itself:
5/29/2014 Once dethrone triggers, it doesn’t matter what happens to the players’ life totals before the ability resolves. You’ll put a +1/+1 counter on the creature even if the defending player doesn’t have the most life as the ability resolves.
The exception to this is the "intervening if" clause, which is specialized rules text in the form "When/Whenever/At [trigger event], if [condition], [effect]."
Yes, cutthroat will get a counter.
The ability is triggered by "whenever this creature attacks the player with the most life or tied for the most life, put a +1/+1 counter on it".
So as soon as the declare attackers phase ends and cutthroat is attacking it will check this condition and if it is true the counter is added.
"In response to the dethrone ability" means that the dethrone trigger is already on the stack and it will resolve after Dismember.
603.2. Whenever a game event or game state matches a triggered ability’s trigger event, that ability automatically triggers. The ability doesn’t do anything at this point.