Can an indestructible creature die by a combination of damage and -X/-X effects?

Question

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?

Answer 1

Say I have a 5/5 indestructible creature. An opponent plays a 3 damage spell on this creature. After that spell resolves,

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.)

the opponent then plays a spell which gives the creature -2/-2 until end of turn.

So it becomes a 3/3 with 3 damage marked on it.

Does the creature die?

No. It has positive toughness, and lethal damage does not destroy indestructible creatures.

700.4. If a permanent is indestructible, rules and effects can't destroy it. (See rule 701.6, "Destroy.") Such permanents are not destroyed by lethal damage, and they ignore the lethal-damage state-based action (see rule 704.5g). Rules or effects may cause an indestructible permanent to be sacrificed, put into a graveyard, or exiled.

Answer 2

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.

700.4. If a permanent is indestructible, rules and effects can't destroy it. (See rule 701.6, "Destroy.") Such permanents are not destroyed by lethal damage, and they ignore the lethal-damage state-based action (see rule 704.5g). Rules or effects may cause an indestructible permanent to be sacrificed, put into a graveyard, or exiled.