Does the Hawl (zakat year) begin on a specific day of the Hijri year or from the moment one possesses the nisab (minimum wealth subject to zakat)? And is zakat due for another Hawl on the same amount if $4875 remains after its zakat was paid when it was $5000?

Regarding Zakāt on wealth, it begins from the day one takes possession of the niṣāb, and Zakāt must be paid when a full year has passed over it, which is a niṣāb in lunar months. It is not a condition for the wealth to increase during the year; rather, Zakāt is to be paid from it as long as it has not fallen below the niṣāb.

The threshold of the niṣāb is the equivalent value of 85 grams of gold, or 595 grams of silver, and the amount to be paid is a quarter of a tenth (2.5%). If the day for the completion of the year arrives and the remaining amount is a niṣāb, this percentage must be paid, whether it decreased during the year, remained constant, or increased; this is because Zakāt is tied to the wealth as long as it constitutes a niṣāb. If it falls below the niṣāb at the completion of the year, then there is no Zakāt on it.