Short answers, pulled from the story.
Seyyed Hossein Nasr and Henry Corbin coined the name School of Isfahan for this philosophical tradition. Mir Damad founded the school through his unique approach to knowledge while serving as a central figure in its early development.
Shah Abbas of the Safavid dynasty turned his attention to intellectual traditions in Islam during the early 17th century. This political focus made Isfahan a famous academic city alongside Ray and Shiraz during that period.
Mir Damad's theory on Time became popular enough to be known as Huduth Dahri. Mulla Sadra later criticized this specific philosophical opinion regarding time despite being Mir Damad's pupil.
Several philosophers continued traditions established by the three primary masters including Sayyed Ahmad Alavi and Shams Addin Muhammad Gilani. Mohsen Fayz Kashani and Mirza Rafiaa Naeini expanded upon ideas introduced by Shaykh-i Baha'i in later decades.
According to Nasr the school plays an important role regarding relations between philosophy and prophecy. It also served critical functions in training future generations of thinkers including Mulla Sadra himself.