Is the jewelry presented to the bride, the value of which was determined by agreement with her father, considered part of the Mahr (dowry) or a gift? And is it a condition for the Mahr to be money, or can it be gold or something else?

The "shabakah" (engagement gift) can be a gift or a mahr (dowry), depending on custom. What was offered to the woman in the question is a mahr because it was negotiated and agreed upon. There is no specific type of wealth required for the mahr; rather, it must be valuable, pure, usable, deliverable, and known. Everything that is valid as a sale can be valid as a mahr. A man should fear Allah and not be excessive in the mahr, for among the blessings of a woman is the facilitation of her engagement and her mahr.