How sound and lawful is the Mudarabah process with the steps mentioned, what is the ruling on one who has previously dealt with it, and what should he do?

The transaction described in the question – where the bank gives the client a check for the car's amount and then takes installments from him with added interest – is unlawful and impermissible, because it is an interest-based loan (riba), not a sale. This is because the bank does not buy the car nor take possession of it before selling it to the client.

For the purchase of goods through banks to be permissible, two conditions must be met: 1. The bank must own the item before selling it. 2. The bank must take possession of the item and transport it before selling it to the client. If one or both of these conditions are missing, the transaction is prohibited. This is based on the Prophet Muhammad’s (peace be upon him) prohibition of selling goods before taking possession of them.

As for someone who engaged in this transaction in the past, ignorant of its prohibition, there is no sin upon them, based on the Almighty’s saying: ﴿So whoever receives an admonition from his Lord and stops, then for him is what has already passed﴾ (Al-Baqarah: 275). However, whoever knows of its prohibition and still engages in it has committed a grave sin and must repent and resolve not to return to it. There is no harm in their continued use of the car after repentance.