What is the ruling on my cashing out the remaining balance of my friend's car to the institution, and the transfer of its ownership to me until the amount is paid in installments with a known profit increase, especially since my friend does not yet have papers proving his ownership of the car? And is it necessary to transfer the car's ownership to me and give him a power of attorney from me, or is it permissible to transfer it directly to him from the institution and suffice with written guarantees?

If your friend bought a car by installments, it is legally his property unless the contract stipulates that ownership does not transfer until all installments are paid. This is a void condition that invalidates the sale from its very foundation. Therefore, your payment of money to the institution has no meaning other than usury, because you are lending your friend what he owes to be repaid with an increase. If your friend's contract is invalid, then the contract is annulled and his money is returned, then you can buy the car from the institution and sell it to him by installments.