Is the rent received by the questioner from his company for leasing his car to it permissible or forbidden, knowing that the company agreed to the lease but has not yet used the car, and that it provides him with another car?

If the authority responsible for the company permits the use of the car before a driver is available, then there is no harm in that, and the wage is lawful, because they consented to the car remaining with you, thus making it a gift of usufruct. However, if they did not permit you to use it, and left it as a trust, then you are not allowed to use it. If you do, they have the right to claim a fee for its use.