Can someone Bioshift counters off a creature after combat damage has been assigned?
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.