Casting a spell without paying its mana cost has no effect on its mana value. Mogg Salvage's mana value is always 3.

You don't sacrifice Nevinyrral's Disk to activate its ability. It's destroyed as part of the ability's resolution if it's still on the battlefield. If an effect gives Nevinyrral's Disk indestructible or regenerates it, it stays on the battlefield.

To cast Mogg Salvage without paying its mana cost, any opponent may control an Island, not just the one who controls the target artifact.

Who controls what lands is important only as you cast Mogg Salvage. Once it has been cast, you no longer controlling a Mountain or each opponent no longer controlling any Islands won't affect Mogg Salvage.