Is it permissible to convert the amount of a dollar remittance directly into riyals at the exchange company, or must one receive the dollars and then exchange them hand to hand?

There is no harm in this transaction because the exchange of dollars for riyals fulfills the condition of immediate possession (al-qabd) in the session. The dollars are in the bank teller's (sarraf's) liability, and exchanging something that is in one's liability after it becomes due is like exchanging something that is in hand, due to the Prophet's (peace and blessings be upon him) saying: "There is no harm in taking the price of its day as long as you do not separate while there is still something between you."