A friend has a Primordial Hydra with 10 +1/+1 counters on it, giving it trample. I block it with a Vampire Nighthawk, causing it to die. My friend then uses Bioshift to take the counters off "in response to it taking lethal damage".
A: Can he do this?
B: Would my creature die and I take the remaining damage if he does it?
No he can not do this.
Damage does not use the stack.
510.1. First, the active player announces how each attacking creature assigns its combat damage, then the defending player announces how each blocking creature assigns its combat damage. This turn-based action doesn't use the stack. A player assigns a creature's combat damage according to the following rules:
510.2. Second, all combat damage that's been assigned is dealt simultaneously. This turn-based action doesn't use the stack. No player has the chance to cast spells or activate abilities between the time combat damage is assigned and the time it's dealt. This is a change from previous rules.
Then once the damage is assigned and SBA are checked, IE it goes to the grave yard, then players have priority.
510.3. Third, any abilities that triggered on damage being assigned or dealt go on the stack. (See rule 603, "Handling Triggered Abilities.")
510.4. Fourth, the active player gets priority. Players may cast spells and activate abilities.
So at the time he wants to move the counters the creature would be dead and gone.
EDIT: In response to B:
Since the nighthawk only has a toughness of 3, in step 510.1 your friend would assign 3 damage to the nighthawk and 7 damage to you if the hydra was 10/10 with trample.
When it comes to trample you only ever have to assign lethal damage to the blocker before you assign damage to the player the blocker is blocking.
No.
First of all, one can only do something in response to a spell or ability being added to the stack, so one can't cast Bioshift in response to damage.
Now, let's look at the real question. Is there a time when Bioshift could be cast such that the Hydra will deal full damage without dying? No. Here's how the turn goes:
Bioshift would need to be cast between 3.2/3.3 and 3.4, but players do not get priority then.
This has not always been the case. From 6th Edition (1999) up until Magic 2010 (2009), Combat damage was placed on the stack, and one could respond to it. Item 5 of this article explains the reason for the change.
He has to assign at least 3 damage to the Vampire Nighthawk (since it has three toughness and hasn't yet taken any damage) and up to 10. The difference (presumably 7) would be assigned to you, the attacked player.
It's a little bit of a tangent, but I thought it might be worth adding an answer about what the controller of Primordial Hydra should do if he wants to bioshift his counters onto another creature.
He can't do it in response to damage, as has been expertly explained in other answers, so I wont repeat it here, this answer is mostly about improved strategy. He can kill your nighthawk and save at least most of his counters, he just wont be able to trample over the damage from them this turn.
If he casts bioshift after you have declared blockers, but before damage is dealt, he can transfer all but 3 counters from his hydra onto another creature he controls. This lets him kill the vampire nighthawk but keep most of his counters on another creature.
If the creature that gets the counters is also attacking, then he still gets all the damage, in the instance that it isn't, then at least he keeps most of his counters for later attacks(and a creature with 7 +1/+1 counters on it in addition to anything else it does is pretty sweet to have!)
