Are the following names forbidden: Al-Mu'alij (The Healer/Therapist), Ar-Rahimah (The Merciful Woman), Ar-Rahmah (Mercy), Al-'Azizah (The Mighty Woman)?

The basic ruling is that it is permissible to name with any name, as long as it does not contain a meaning that necessitates its prohibition, such as a name enslaved to other than Allah (e.g., 'Abd al-Lat), or one that includes self-praise, or a foreign name whose meaning is unknown. Therefore, there is no problem in naming with "al-Mu'alij (the Healer), al-Azizah (the Mighty/Dear), and al-Rahimah (the Merciful)." As for "al-Rahmah (Mercy)," it is better to avoid naming with it, due to the meaning for which the Prophet, peace and blessings be upon him, disliked naming with "Rabah (Gain) and Yasar (Ease)."