Why do angels need God's mercy, even though they are obedient, have no free will, and commit no sins? And why is the angel on the right greeted with the addition of "and His mercy and His blessings" over the angel on the left?

A worshiper has multiple intentions when performing the taslim (salutation) at the end of prayer. In addition to the intention of exiting the prayer, the Imam intends to greet the congregants, and the congregants intend to greet the Imam and those with them. As for the one praying alone, he intends to exit the prayer and greet the angels. Scholars have differed regarding the addition of "wa barakatuh" (and His blessings) in the taslim. The majority opinion is to limit it to "Al-salamu alaykum wa rahmatullah" (Peace be upon you and the mercy of Allah). Praying for mercy for the angels is not deemed reprehensible by Islamic law, as they are creatures in need of Allah's mercy and servants obligated to obey. No one enters Paradise except by the mercy of Allah.