Is it obligatory to say "صلى الله عليه وسلم" (may Allah bless him and grant him peace) or "عليه الصلاة والسلام" (upon him be prayers and peace) when mentioning the Messenger, and is it true that Allah does not pray upon the Messenger, but rather the Messenger is the one who prays upon Allah?

Sending blessings upon the Prophet when he is mentioned is recommended, not obligatory. It is permissible in any form, such as "peace and blessings be upon him" or "may Allah bless him and grant him peace." Allah Almighty's blessings upon His Prophet Muhammad, may Allah bless him and grant him peace, mean His praise of him among the exalted assembly of angels.