Are all mosques considered to have a haram (sacred inviolable zone) within which hunting birds is prohibited, or is this matter exclusive to the Sacred Mosque (Masjid al-Haram) and the Prophet's Mosque (Masjid an-Nabawi)?

The prohibition of hunting is specific to the two Holy Sanctuaries (Mecca and Medina), whether inside or outside the mosque. No other places or mosques are to be analogized to them.

Whoever hunts pigeons must avoid polluting the mosque. Therefore, he should throw them outside the mosque or hunt them after leaving it.

A pigeon that has strayed from a specific owner belongs to that owner, and it is not permissible for anyone else to hunt it. Furthermore, whoever hunts pigeons must intend to benefit from eating them, not merely to hunt for amusement.