As Cascabel said in a comment, the phrasing "Can my opponent morph a dead creature" is a bit inappropriate, because the creature isn't dead. What happens is that when Lightning Bolt is cast, it's put on the stack. If it were to resolve at that point, it would deal damage to the creature, and then the creature would die as a State Based Action. However, before it resolves, your opponent gets priority, and if they use their priority to morph their creature at that point, the Lightning Bolt hasn't resolved yet, and the creature hasn't received damage or died yet.
Since morphing doesn't use the stack, the creature would immediately become a 0/5. There would then be another round of priority, so if you have another Lightning Bolt, you could cast it at this time. But if nothing else happens, the Lightning Bolt would then resolve, and by this time the creature is 0/5, and so would not die.