Is it permissible and free of usury to contract for a commodity by leasing it for 60 months, with checks written for the full period, and with the advantage of calculating each month as three months of payment if paid before the first of the month, and one month if paid after, and then gifting the commodity upon completion of payment?

This transaction is one form of Ijara Muntahia Bittamleek (lease ending in ownership), where the Islamic Fiqh Academy has permitted the conjunction of a lease contract with the gifting of the asset to the lessee, conditional upon the full payment of the rent, or a promise of gifting after the full payment of the rent. It is stipulated that the gift contract must be independent of the lease contract, and that the lease must be genuine and not a mere cover-up for a sale, such that the warranty of the goods and non-operational maintenance expenses fall upon the lessor.

As for what you mentioned, "if payment is made before the first day of each month, the month is counted as three months of payment, and if payment is made after that, it counts as one month," this is known as "da' wa ta'ajjal" (forgive and expedite), and scholars have differed on its permissibility. The Islamic Fiqh Academy has ruled in favor of the permissibility of "a reduction from a deferred debt, for the sake of expediting it," provided it is not based on a prior agreement. Accordingly, this specific scenario is not permissible due to the existence of a prior agreement at the time of the contract.