Must the husband spend the same duration of the 'Umrah (15 days) accompanying the second wife in her home, and what are her rights during this period?

If the trip is at the expense of the first wife, then there is no harm for the husband to accompany her without the second, and he is not required to seek permission from the second. However, he is obligated to compensate the second wife for the period he spent with the first wife, unless the second wife waives her right to equal division.