Can I deal damage to Hornet Nest multiple times before it dies?
If it is your turn, you may cast two instances of Lightning Axe (or any other instant) before your opponent gets priority. However, they do not resolve until your opponent has a chance to respond. 

The order is:


You get priority
You cast Lightning Axe, choosing targets and paying the costs; Lightning Axe is on the stack
You intentionally retain priority.
You cast Lightning Axe, choosing targets and paying the costs; second Lightning Axe is on stack.
You get priority, you pass
Opponent gets priority, they may cast a spell.
Opponent passes priority.
Second Lightning Axe checks if the target is still legal. It is, it resolves. Creature has lethal damage.
a. 5 Damage has been dealt to Hornet Nest, so next time priority would be given, ability that creates 5 Hornet tokens would be put on stack.
State-based actions are checked, creature dies. 
Ability that creates 5 Hornet tokens goes on stack. (It resolves)
You get priority, pass
Opponent gets priority, pass.
First Lightning Axe checks if target is still legal.  It is not, so it fizzles. No damage is dealt to Hornet Nest


Final result: 5 Hornet tokens