If Veiling Oddity is suspended, then when the last time counter is removed from it, both its triggered ability and the “cast this card” part of the suspend ability will trigger. They can be put on the stack in either order.

Regardless of which order those abilities resolve in, Veiling Oddity can’t be blocked that turn. The ability affects all creatures, not just the creatures that were on the battlefield when it resolved.

If Veiling Oddity is suspended and the last time counter is removed during another player’s turn, that player’s creatures can’t be blocked that turn, too.

You can exile a card in your hand using suspend any time you could cast that card. Consider its card type, any effect that affects when you could cast it (such as flash) and any other effects that could stop you from casting it (such as Meddling Mage’s effect) to determine if and when you can do this. Whether or not 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 requires a target even if no legal targets are available at that time.

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

When the last time counter is removed, the second triggered ability of suspend will trigger. It doesn’t matter why the last time counter was removed or what effect removed it.

If the second triggered ability of suspend (the one that lets you cast the card) is countered, the card can’t be cast. It remains exiled with no time counters on it, and it’s no longer suspended.

As the second triggered ability resolves, you must cast the card if able. Timing restrictions based on the card’s type are ignored.

If you can’t cast the card, perhaps because there are no legal targets available, it remains exiled with no time counters on it, and it’s no longer suspended.

If the spell has any mandatory additional costs, you must pay those if able. However, if an additional cost includes a mana payment, you are forced to pay that cost only if there’s enough mana in your mana pool at the time you cast the spell. You aren’t forced to activate any mana abilities, although you may do so if you wish.

A creature cast using suspend will enter the battlefield with haste. It will have haste until another player gains control of it (or, in some rare cases, gains control of the creature spell itself).