The Tenth Doctor

If the first triggered ability of suspend (the one that removes time counters) is countered, no time counter is removed. The ability will trigger again at the beginning of the card's owner's next upkeep.

If you cast a card "without paying its mana cost," such as with suspend, you can't choose to cast it for any alternative costs. You can, however, pay additional costs. If the card has any mandatory additional costs, you must pay those if you want to cast the card.

If the second triggered ability is countered, the card can't be cast. It remains exiled with no time counters on it, and it's no longer suspended.

You can exile a card in your hand using suspend any time you could cast that card. Consider its card type, any effects that modify when you could cast it (such as flash) and any other effects that stop you from casting it (such as from Meddling Mage's ability) to determine if and when you can do this. Whether you could actually complete all steps in casting the card is irrelevant. For example, you can exile a card with suspend that has no mana cost or that requires a target even if no legal targets are available at that time.

If the spell requires any targets, those targets are chosen when the spell is finally cast, not when it's exiled.

Cards exiled with suspend are exiled face up.