How long can the spells exiled with Lukka, Coppercoat Outcast be cast?
You can cast a card exiled with Lukka until the end of the game, but only once. As soon as you cast the card (or it leaves exile for any other reason), it loses that effect.
Lukka's ability creates a continuous effect that modifies the game rules for the creature cards it exiles. As you noted, it has no duration, so it lasts until the end of the game.

611.2a A continuous effect generated by the resolution of a spell or ability lasts as long as stated by the spell or ability creating it (such as “until end of turn”). If no duration is stated, it lasts until the end of the game.

However, once you do cast one of these cards, it moves from exile to the stack. In that moment, it becomes a new object that no longer has that ability:

400.7. An object that moves from one zone to another becomes a new object with no memory of, or relation to, its previous existence. [..]

Likewise, if that creature is later exiled through other means, for example Eat to Extinction, it will again become a new object, and you cannot cast it again through Lukka even though it is exiled.