What is the reward for cleaning the mosque, especially for cleaning the washrooms at all times, and does the doer of that receive the reward stipulated for it in the Quran and Sunnah?

Cleaning the mosque is a desirable act, as indicated by various hadiths, including the Prophet's ﷺ command to build mosques in dwellings, to clean them, and to perfume them. This is also supported by the actions of the Companions and the early Muslims, such as Abdullah, who used to cense the mosque, and Ibn Al-Zubayr, who perfumed the Kaaba with musk.

As for cleaning the facilities attached to the mosque, a Muslim is rewarded for it because it falls under serving and benefiting the mosque's users. This is a righteous deed that falls under the general meaning of Allah's Almighty saying: (So whoever does an atom's weight of good will see it).