Creatures put onto the battlefield tapped don’t cause Quest for Renewal’s first ability to trigger.

Controlling more than one Quest for Renewal with four or more quest counters on it is redundant. You can’t untap your permanents more than once in a single untap step.

During another player’s untap step, effects that would otherwise cause your creatures to stay tapped don’t apply because they apply only during *your* untap step. For example, if you control a Quest for Renewal with four or more quest counters on it and a Deep-Slumber Titan (a creature that says “Deep-Slumber Titan doesn’t untap during your untap step”), you untap Deep-Slumber Titan during each other player’s untap step.

As another player’s untap step begins, if there are four or more quest counters on Quest for Renewal, all your creatures untap during that untap step. You have no choice about what untaps. Those creatures untap at the same time as the active player’s permanents.