What is the ruling on repeating a specific dhikr, such as Sayyid al-Istighfar an unlimited number of times before the Fajr athan, and the complete Salawat al-Ibrahimiyyah an unlimited number of times only on Thursday night and Friday?

Designating a specific time for worship is permissible if it is for the individual's availability and vigor, not as an act of worship itself or an اعتقاد of its specific virtue. Seeking forgiveness (Istighfar) during the pre-dawn hours (Sahar) before Fajr is commendable, a virtuous and prescribed act. Allah has praised those who seek forgiveness in the pre-dawn hours. The "Sayyid al-Istighfar" (Master of Seeking Forgiveness) is a virtuous formula, and one can vary between it and other formulas. It is sunnah to increase salutations upon the Prophet, peace and blessings be upon him, on Friday and its eve. The Ibrahimiyyah salutation is the best form of salutation upon the Prophet, and choosing it for that virtuous time is prescribed.