Can you use imprint abilities on cards like Isochron Scepter to exile a card with X in its mana cost and cast it for free?

Yes, you can exile a card with X in its mana cost using Isochron Scepter's imprint ability, provided the card's mana value is 2 or less (with X treated as 0 for that check, per CR 202.3b). So for example, a card like Brainstorm (CMC 1) can be imprinted, but a card like Fireball (with X in its cost) would also have its mana value calculated with X=0, making it eligible if its non-X portion is 2 or less.

When Isochron Scepter's activated ability lets you cast the imprinted card 'without paying its mana cost,' you cast it for free. However, CR 107.3b states that if a spell has X in its mana cost and is cast without paying that cost, the value of X is 0. You cannot choose a different value for X because you are not actually paying mana for the spell.

This means the spell resolves with X=0. For many X spells this produces a minimal or useless effect — a Fireball with X=0 deals 0 damage, for instance. The spell still resolves and you still get any non-X effects, but X will always be zero in this scenario.

Concrete example: You imprint Grapeshot (storm, CMC 2) — wait, bad example. Say you imprint Brainstorm, that works fine. If instead you imprint Fireball (X+R, mana value = 1 with X=0), you can activate Scepter and cast Fireball for free, but X=0, so it deals 0 damage to the chosen target — a legal but useless cast.

References: CR 202.3b (X=0 for mana value calculations), CR 107.3b (X=0 when casting without paying mana cost), and Isochron Scepter's own Oracle text restricting imprint to instants with mana value 2 or less.