Is the prayer room (musalla) inside the building considered a mosque where prayer is obligatory, or is it the independent prayer room (musalla) 300 meters away, or both? And which of them is more deserving of prayer, given the presence of bid'ahs (innovations) in the second prayer room (musalla)?

Prayer is valid in both mosques. The mosque built under a house is valid. Additional innovations do not prevent prayer in a mosque. It is preferable to pray in the mosque of the neighborhood, then in the one with the larger congregation. However, if devotion is achieved in one of the two mosques, that one is preferable for prayer, while advising the people of the mosque that deviates to follow the Sunnah.

If the Imam is an innovator, it is better to pray in the other mosque, because praying behind a transgressor (fasiq) or an innovator (mubtadi') is disliked (makrooh).

If one of the places is merely a prayer area (musalla) and not a mosque, then praying in an endowed mosque (masjid mawquf) is preferable. Mosques have special rulings, including: the obligation of "Tahiyyat al-Masjid" (salutation to the mosque), the prohibition of those in a state of major ritual impurity (junub) and menstruating women from entering except with ablution, and the unlawfulness of buying and selling within it.