Is zakat obligatory on the down payment from the sale of the old apartment, whether or not it was used to pay off part of the debt for the new apartment? And is zakat obligatory on the monthly installments from the sale of the old apartment, and what is the mechanism for its zakat?

The debt you owe the bank does not necessitate zakat on its equivalent amount from your wealth, unless you possess additional wealth exceeding your needs, in which case it is offset against the debt, and you then pay zakat on all your wealth. However, if you are cautious and pay zakat on your entire wealth, it is more absolving for your conscience. If you use the amount received from the apartment sale to repay the debt, then no zakat is due on that specific amount.

As for the installments due to you from the apartment buyer, the preponderant opinion is that zakat is obligatory upon the creditor for them, without exception. The zakat on these installments is deferred until they are collected. Once collected, you pay zakat for the past years since the apartment was sold. So, if you collect an installment at the end of the first year, you pay zakat for one year. If you collect it at the end of the second year, you pay zakat for two years, and so on. You are not obligated to pay zakat on the remaining installments until they are collected, and this is after deducting the debt you owe to the bank.