The delayed triggered ability will trigger whether Forger's Foundry is still on the battlefield or not.

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.

If a 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.

When casting an instant or sorcery spell with in its mana cost using the mana added by Forger's Foundry, use the value of X chosen when you cast that spell to determine whether or not the delayed triggered ability triggers.

The mana added by Forger's Foundry can be spent on anything, not just instant and sorcery spells with mana value 3 or less.

You choose which spells (if any) to cast as Forger's Foundry's last ability resolves. If you do, you do so as part of the resolution of that ability. You can't wait to cast them later in the turn.

You cast the spells one at a time as Forger's Foundry's last ability is resolving, choosing modes, targets, and so on. The last spell you cast will be the first one to resolve. Ignore timing restrictions based on the cards' types. Other timing restrictions, such as "Cast [this spell] only during combat," must be followed.