Skithiryx can regenerate even if it isn't in combat, it's already tapped, or it's undamaged.

Damage that a creature with infect deals doesn't result in damage being marked on a creature or a player losing life. Instead, it results in that many -1/-1 counters being put on that creature or that many poison counters being given to that player. Damage dealt to planeswalkers still results in that planeswalker losing that many loyalty counters. Damage dealt to battles still results in that battle losing that many defense counters.

A player who has ten or more poison counters loses the game. This is a state-based action.

The -1/-1 counters remain on the creature indefinitely. They're not removed if the creature regenerates or the turn ends.

Damage from a source with infect is damage in all respects. If the source with infect also has lifelink, damage dealt by that source also causes its controller to gain that much life. Damage from a source with infect can be prevented or redirected. Abilities that trigger on damage being dealt will trigger if a source with infect deals damage, if appropriate.

Activating the regeneration ability creates a replacement effect that acts like a shield, replacing the next time Skithiryx would be destroyed that turn. The shield works against effects that try to destroy Skithiryx or lethal damage that would be dealt to Skithiryx.

If the regeneration shield is used, Skithiryx isn't destroyed, all damage marked on Skithiryx is healed, it becomes tapped, and it's removed from combat (if applicable).