Is it permissible to give zakat to a sister who is married to a husband unemployed due to his engagement in da'wah (Islamic proselytizing), which renders them in need, and does this discourage him from earning a living?

It is permissible to give an needy sister from zakat funds, especially if her husband does not provide her with sufficient support. She has priority in receiving zakat, as it is both charity (sadaqah) and maintaining family ties (silah). The fact that the husband is able to earn prevents him from being eligible for zakat, but it does not prevent the poor wife. An able-bodied husband is advised to work to maintain his own dignity and that of his family, and this is better than being preoccupied with worship over earning a livelihood if he is in need.