As Eyes Everywhere’s last ability resolves, Eyes Everywhere must be on the battlefield and the target nonland permanent must be a legal target. If either of these things isn’t true, the ability does nothing.

In a multiplayer game, if a player leaves the game, all cards that player owns leave as well. If you leave the game, the effect giving you control of the target nonland permanent ends.

The effect of Eyes Everywhere’s last ability lasts indefinitely. It doesn’t wear off during the cleanup step, and it doesn’t expire if Eyes Everywhere leaves the battlefield.