Can a 0/0 creature with +1/+1 counters on it (like a Hydra) be regenerated?
Those two spells Destroy. Regeneration replaces destruction, which means the destruction doesn't happen, which means it doesn't cause anything to go to the graveyard. If you cast the regeneration mode of Golgari Charm in response to Doom Blade or Supreme Verdict, your Hydras will be safe. They will not lose counters, enchantments, or equipment; they don't gain summoning sickness; spells targeting them don't lose sight of them; LTB and ETB abilities don't trigger; etc.


  614.8. Regeneration is a destruction-replacement effect. The word “instead” doesn’t appear on the card but is implicit in the definition of regeneration. “Regenerate [permanent]” means “The next time [permanent] would be destroyed this turn, instead remove all damage marked on it and tap it. If it’s an attacking or blocking creature, remove it from combat.” Abilities that trigger from damage being dealt still trigger even if the permanent regenerates. See rule 701.12.


Regeneration will not save the Hydras from being sent to the graveyard as a result of having non-positive toughness, but nothing in the scenario you gave reduces their toughness (by removing +1/+1 counters or otherwise).