Does using a free ticket count for one of my four qualifying transactions?

No, you must pay for at least part of the price of one ticket in an order for it to count as a qualifying transaction toward earning a Free Reward Ticket.

The Atom Rewards introductory offer of “See 4 movies, earn a free standard ticket” ends on December 31, 2018. Only members who saw their first qualifying movie on or before December 31, 2018 are eligible to earn a free movie ticket under the Atom Rewards offer.