The mana value of a spell cast without paying its mana cost is determined by its mana cost, even though that cost wasn't paid.

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 card has {X} in its mana cost, you must choose 0 as the value of X when casting it without paying its mana cost.

Suspend is a keyword that represents three abilities. The first is a static ability that allows you to exile the card from your hand with the specified number of time counters (the number before the dash) on it by paying its suspend cost (listed after the dash). The second is a triggered ability that removes a time counter from the suspended card at the beginning of each of your upkeeps. The third is a triggered ability that gives you the option to cast the card when the last time counter is removed.

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.

Cards exiled with suspend are exiled face up.

Exiling a card with suspend isn't casting that card. This action doesn't use the stack and can't be responded to.

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

If an effect refers to a "suspended card," that means a card that (1) has suspend, (2) is in exile, and (3) has one or more time counters on it.

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.

When the last time counter is removed, the second triggered ability of suspend (the one that lets you cast the card) triggers. It doesn't matter why the last time counter was removed or what effect removed it.

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.

Due to a recent rules change to suspend, you are no longer required to cast the suspended card as the second triggered ability of suspend resolves. Instead, as the second triggered ability resolves, you may cast the card. Timing permissions based on the card's type are ignored. If you don't cast the card, it remains exiled with no time counters on it, and it's no longer suspended.

Time counters are usually found on cards with suspend and vanishing, but may be found on other cards as well. Notably, Sagas use lore counters to track their progress, not time counters. You can't move a Saga's chapters forward and backward this way.

To time travel, look at each permanent you control with a time counter on it and each suspended card you own in exile with a time counter on it. For each of them, you choose whether you want to put a time counter on that card or permanent, remove a time counter from it, or do neither. Then those changes all happen simultaneously.