Is it permissible to pray in congregation in the mosque of a private residential complex, and will the full reward for congregational prayer be counted, given that there is a more distant neighborhood mosque and strangers are prohibited from entering the complex?

If the designated prayer area is not an endowed mosque, then it is a musalla (prayer place) where congregational prayers are valid and sufficient, thus fulfilling the obligation. However, it is known that praying in an endowed mosque is superior and yields greater reward. A mosque is not a prerequisite for congregational prayer; rather, it is permissible in any place where prayer is valid.