If you exile a card as the triggered ability resolves, that card will remain exiled whether you cast the copy or not.

Casting the copy happens as part of the resolution of Demilich's triggered ability. If you choose to cast it, you must do so right away. If you choose not to cast it right away, the copy ceases to exist the next time a player would receive priority.

While casting Demilich, if you are required to pay an additional cost that includes a generic mana component, Demilich's cost reduction effect can apply to it. For example, if you're required to pay an additional to cast Demilich (perhaps because of Sphere of Resistance), and you've cast five or more instant and/or sorcery spells this turn, you would pay to cast Demilich.

You must pay all normal costs for casting the copy.

Demilich's last ability doesn't change when you can cast it. In most cases, you can cast Demilich only during your main phase.