As Magosi's third ability resolves, you'll skip your next turn even if you don't put an eon counter on Magosi (because it's left the battlefield by then, perhaps).

You can activate Magosi's fourth ability only if it has an eon counter on it.

You return Magosi to its owner's hand as part of the cost to activate its fourth ability. If Magosi has more than one eon counter on it, you can't activate the ability, untap it, then activate it again.

If you somehow activate Magosi's third and fourth abilities during the same turn, the effects will cancel each other out. The turn you skip is the next one you would take, which is the turn created by Magosi's fourth ability. It doesn't matter in which order the abilities resolved.

If you take an extra turn in a Two-Headed Giant game, you entire team takes the extra turn.