Is it permissible to perform more than one Sunnah in a single prayer, such as saying "Rabbana wa laka al-hamd" (Our Lord, and to You be all praise) in one rak'ah, and "Allahumma Rabbana wa laka al-hamd" (O Allah, our Lord, and to You be all praise) in another rak'ah, and similarly in bowing (ruku') and prostration (sujud)?

Yes, it is recommended for the worshipper to use all the confirmed formulas from the Prophet, peace and blessings be upon him, and to vary between them. Sheikh Al-Uthaymeen mentioned that the variation in acts of worship and remembrances is a blessing from Allah due to the benefits it brings: presence of heart, choosing what is easier and simpler, and increasing praise of Allah. Thus, it is permissible to perform more than one Sunnah for a single action in prayer to increase vitality and devotion, and so that the Sunnahs are not lost.