Say I have a 5/5 indestructible creature. An opponent plays a 3 damage spell on this creature. After that spell resolves, the opponent then plays a spell which gives the creature -2/-2 until end of turn. Does the creature die?
It becomes a 5/5 with 3 damage marked on it.
Perhaps you've played DotP where this is shown as 5/2 on the screen, but that's misleading because damage does not reduce toughness. (Damage from a source with Wither or Infect can grant -1/-1 counters which in turn reduce toughness, but that doesn't appear to be what you're asking about.)
So it becomes a 3/3 with 3 damage marked on it.
No. It has positive toughness, and lethal damage does not destroy indestructible creatures.
No, he would be a 3/3 marked with 3 damage. This 3 damage would be 'lethal', but since he is indestructible, he is immune to such deaths.