Goliath Daydreamer's last ability allows you to cast an exiled card with a dream counter on it only while the ability is resolving and still on the stack. You can't wait to cast the spell later in the turn. Timing restrictions based on the card's type are ignored.
If a spell is countered or otherwise fails to resolve, Goliath Daydreamer's first ability won't exile it.
If the spell you cast has {X} in its mana cost, you must choose 0 as the value of X when casting it without paying its mana cost.
Magic: The Gathering · ™ & © Wizards of the Coast · Illus. Omar Rayyan
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The spell will have all of its normal effects before being exiled. If the spell's normal effects include exiling itself, the effect of Goliath Daydreamer's first ability won't apply to it.
Since you are using an alternative cost to cast the spell, you can't pay any other alternative costs. You can, however, pay additional costs, such as kicker costs. If the card has any mandatory additional costs, you must pay those.
