Is the process of purchasing US dollars with Indonesian Rupiah—where the agreement is made in Yemen and the Rupiah is sent from Indonesia an hour or hours later, after which the merchant sends the dollars later via exchange offices—considered usury, and does constructive possession apply therein? What is the correct Shar'i method for completing this contract?

The contract you mentioned does not fulfill the condition of immediate possession (taqabud), neither actually nor constructively. This is because you entered into a currency exchange (sarf) contract, and the merchant did not transfer the dollars until hours later. In a sarf contract, immediate possession must be realized, either actually or constructively; otherwise, it is invalid. Jurists have reached a consensus that if one currency is exchanged for another of the same type or for a different type, immediate possession must occur in the same session; otherwise, the contract is null and void.