Does a creature with regeneration survive being dealt lethal damage and destruction in the same combat?

Regeneration works by setting up a 'regeneration shield' on the creature (CR 701.15). When a regeneration shield is expended, it does three things simultaneously: it removes all damage marked on the creature, taps it, and removes it from combat — all instead of being destroyed (CR 701.15b).

Lethal damage causes a creature to be destroyed via state-based actions (CR 704.5g). Destruction effects also destroy. Crucially, one regeneration shield handles one destruction event, regardless of whether that event comes from lethal damage, a spell, or a combat effect. If both lethal damage and a destruction effect would trigger in the same moment, only one shield is consumed to replace that single destruction (CR 701.15b).

However, if a creature would be destroyed twice from separate independent events and you only have one shield, the shield saves it from the first destruction, but you would need a second shield for any subsequent one. Also, regeneration cannot save a creature from being 'exiled' or effects that say 'cannot be regenerated' (CR 701.15d).

Example: Your 2/2 has regeneration active. In combat, it blocks a 3/3 (lethal damage) and your opponent also activates an ability saying 'destroy target creature' in the same combat damage step. State-based actions check: the creature has lethal damage and a destroy effect pending. The regeneration shield fires once — the creature survives, is tapped, and removed from combat.

So yes, one regeneration shield is sufficient to survive lethal combat damage and destruction together, as long as the regeneration ability was activated before state-based actions are checked (CR 704.3).