When you activate an ability, you untap the creature with that ability as a cost. The untap can’t be responded to. (The actual ability can be responded to, of course.)

If the permanent is already untapped, you can’t activate its ability. That’s because you can’t pay the “Untap this permanent” cost.

If you don’t play the exiled card that turn, it remains exiled but you can’t play it.

The exiled card is played using the normal timing rules for its card type, as well as any other applicable restrictions such as “Cast [this card] only during combat.” For example, you can’t play the card during an opponent’s turn unless it’s an instant or has flash. Similarly, if the exiled card is a land, you can’t play it if you’ve already played a land that turn. If it’s a nonland card, you’ll have to pay its mana cost. The only thing that’s different is you’re playing it from the Exile zone.

If a creature with an ability hasn’t been under your control since your most recent turn began, you can’t activate that ability, unless the creature has haste.