Is it permissible to pray over the deceased without being certain of their Islam, especially if the mosque prays over everyone who uttered the Two Testimonies, even if they never prayed at all?

The fundamental principle is to offer the funeral prayer (Salat al-Janazah) for every Muslim who pronounces "La ilaha illa Allah" (there is no god but Allah), unless it is known that they committed an act that nullifies their Islam. The one who abandons prayer is considered a disbeliever according to the soundest opinions of scholars. If it is known that the deceased abandoned prayer, then it is not permissible to offer the funeral prayer for them, based on the words of Allah Almighty: (And never offer a prayer for anyone of them who has died, nor stand by his grave). If you are in doubt about the state of the deceased, then the funeral prayer should be offered for them, with a conditional supplication, such as: "O Allah, if he was a believer, then forgive him and have mercy on him." This is what Sheikh Ibn Uthaymeen stipulated, citing as evidence the verse of li'an (mutual imprecation), the hadith of the leper, the bald man, and the blind man, the Istikhara supplication (prayer for guidance), and Ibn Taymiyyah's saying: "You must include the condition, O Ahmad." However, if it is known with certainty that the person did not pray, then it is not permissible to offer the funeral prayer for them.