If I seek a khulʿ (divorce initiated by the wife), will I be required to pay back the mahr (dowry)? And if I remarry, will custody go to the maternal grandmother?

It is not a condition for khulʿ (divorce at the wife's instance) that its compensation be the entire mahr (dowry); rather, it is valid with less or more, depending on the agreement. The custody of the child belongs to the divorced mother unless there is an impediment. If the mother marries a non-mahram, custody transfers to her mother. If her mother is not available, then to other relatives, such as the paternal grandmother or the maternal aunt, or the father, with scholarly disagreement regarding the order of priority. The reference point for disputes regarding custody is the Sharia court.