Does supplication in prayer (such as during prostration and after the final Tashahhud) require praising Allah and invoking blessings upon the Messenger—may Allah's peace and blessings be upon him—before and after it in order for it to be accepted?

Praising Allah and sending blessings upon His Prophet are not obligatory in supplication (Du'a). Rather, they are among the recommended etiquettes with which one begins a supplication, not ends it, whether the supplication is within Salah (definition) (prayer) or outside of it. This is based on the Hadith (definition) of Fudalah ibn 'Ubayd: "When one of you prays, let him begin by praising his Lord and extolling Him, then let him send blessings upon the Prophet (peace and blessings be upon him), then let him supplicate thereafter for whatever he wishes." An exception to this are some instances where supplication is mentioned without praise or blessings, such as the opening supplication of Salah (Du'a al-Istiftah) or the supplication between the two prostrations.