Is it permissible to pray outside the mosque and sit waiting for the sermon, while there are empty spaces inside the mosque, in order to avoid the crowded stairs and to depart quickly?

If the mosque is not full, worshippers must pray inside it, and this includes the upper floors and other areas. Praying outside the mosque is permissible only if it is full or if the rows are connected; overcrowding is not an excuse. However, if they pray in this place, their prayer is valid, even if there is an empty space inside the mosque. It is mentioned in "Al-Taj wal-Iklil" that Malik said it is permissible to pray Friday prayer in the courtyards and precincts of the mosque and in the shops and houses adjacent to it, even if the rows are not connected and without permission, and the prayer of those who pray there is complete if the mosque is full. Ibn Rushd mentioned that whoever prays on the platforms of shops without necessity or a full mosque has done wrong, yet their prayer is valid.