What is the amount on which Zakat is obligatory: is it the amount remaining at the end of the year, or the amount that I had at the beginning of the year before paying installments? And is Zakat obligatory on gold designated for engagement that has been in one's possession for a full year, or is there no Zakat on it?

Installments that you pay during the year are not subject to zakat. However, what remains with you after the completion of the year, if it reaches the nisab (approximately 85 grams of gold), then zakat is due on it. It is more cautious not to deduct the debt from it. The more probable opinion is that if you possess assets that you do not need, they can be used to offset the debt. In that case, you would use them to offset the debt and pay zakat on what you have in hand. Otherwise, you may deduct the debt you owe from the money you possess. If the debt encompasses all that you own, then no zakat is due on you. As for the gold you possess, zakat is due on it if a full lunar year has passed and it reaches the nisab (approximately 85 grams of pure gold), and its zakat is a quarter of a tenth (2.5%).