Is it permissible for a brother to authorize his brother to sell his car on his behalf and pay off his debt to the eldest brother, and is it permissible not to give him the remaining amount from the car’s price in exchange for his contribution to family expenses?

There is no objection to buying the car from him, or paying off the elder brother's debt on his behalf if he authorizes you, or considering the remaining amount of the car's price as a contribution to family expenses. All of this is conditional upon mutual agreement between the two of you. If an agreement is reached, then you are a trustworthy agent and it is not permissible for you to exceed what you have been authorized to do.