Can a creature be killed by damage from two different instants?
Yes, that was a legal move.

You may cast multiple instants in the same turn. There's no limit beyond having the instants and being able to pay their cost.

After taking the first two damage, your 4/4 would still be a 4/4. Those two numbers represent power/toughness - toughness is not health, but the amount of damage a creature can withstand in one turn.

Your 4/4 would be a 4/4 with two damage marked against it after the first spell, then a 4/4 with four damaged marked against it after the second. Since it has taken damage at least equal to its toughness in one turn, it dies.