Is custody the right of the mother or the father in the event that the wife requests a divorce from her unemployed husband, given the father's bad criminal record?

If your sister is divorced from her husband, the custody of her young children belongs to her, not to their father, unless she has an impediment to custody. The mother has the primary right to custody over others, by agreement of the scholars. In disputes concerning custody matters, recourse should be made to the Sharia court for resolution.