Is zakat obligatory on money saved in the bank and on the amount with which I purchased a car, and is the completion of a full year (hawl) a condition for paying zakat on them both?

You should pay the Zakat on your money in the next hawl (lunar year) from the day you received it from selling the land. This is because the land was not for trade, and the money it was sold for had not yet completed a hawl. Newly acquired money, not from trade profit or animal offspring, begins a new hawl. Jurists have likened this to the price of possessions such as real estate. As for the amount you used to buy the car, there is no Zakat on it because it had not completed a hawl before its purchase. There is no Zakat on the car itself unless it is used for hire; in that case, Zakat becomes due on the money it accrues if it completes a hawl while reaching the nisab (minimum threshold) by itself or combined with other money or trade goods.