Am I obligated to pay zakat on the money I received from selling my old car after I had already paid zakat on my original wealth and used it to buy a new car, and is it permissible to give a portion of zakat for the expansion of a mosque with the intention that it is for the sake of Allah?

Zakat (definition) is not obligatory on wealth unless it reaches the nisab (definition) (minimum threshold) and a full Hijri year has passed over it. If the money from selling an old car and buying a new one with it is spent before a year passes, then there is no Zakat due on it. Paying Zakat for the expansion of a mosque is not permissible, as the majority of scholars believe that the eight categories for Zakat distribution, including "in the cause of Allah," are specifically for Jihad.