Is it permissible to supplicate by mentioning a specific person's name during prayer, and is it obligatory to send blessings upon the Prophet when hearing his name during prayer?

It is permissible to supplicate for a person by name or against them in prayer. This is indicated by the Prophet's (peace be upon him) supplication in his Qunut for individuals by their names, and his supplication against Ra'l and Dhakwan. Furthermore, it is recommended for anyone who hears or reads the name of the Prophet (peace be upon him) in prayer to invoke blessings upon him by saying: "صلى الله عليه وسلم" (peace be upon him), and not "اللهم صل على محمد" (O Allah, send blessings upon Muhammad). This is the view held by the Shafi'is, and it is supported by narrations from the Salaf.