If you cast a spell “without paying its mana cost,” you can’t choose to cast it for any alternative costs. You can, however, pay additional costs, such as kicker costs. If the spell has any mandatory additional costs, those must be paid to cast the spell.

Spells that can’t be countered can still be exiled by Smirking Spelljacker’s triggered ability. They won’t resolve.

If the spell you cast has in its mana cost, you must choose 0 as the value of X when casting it without paying its mana cost.

You choose whether to cast the exiled card as Smirking Spelljacker’s last ability resolves. If you do, you do so as part of the resolution of that ability. You can’t wait to cast it later in the turn. Timing restrictions based on the card’s types are ignored.

If you cast a card exiled with Smirking Spelljacker, it will go to the stack. From there, it will likely go to the battlefield or back to its owner’s graveyard. Even if it returns to exile, it will no longer be exiled with Smirking Spelljacker, so you can’t cast it again.