What is the ruling on praying in a mosque that places the name of Allah the Almighty opposite the name of the Messenger of Allah, peace and blessings be upon him, and is the worshipper sinful for praying therein? And is it better to pray Tarawih in a mosque that does not complete the Qur’an, or to pray it at home with the completion of the Qur’an?

Your prayer in the mosque is valid. Placing the Prophet's name with Allah's name is not appropriate, as it may suggest polytheism (shirk), but it does not affect the validity of the prayer. Performing Tarawih prayer in the mosque is better than at home, due to the reward associated with praying with the Imam until he departs. Jurists have differed regarding the preference between praying it in the mosque or at home. Some have said it is preferable to pray it individually at home if it does not lead to the abandonment of mosques. Those who advocate for its superiority in the mosque did not stipulate the completion of the Quran or prolonged standing in prayer.