1. In his mother was raised a Catholic, the phrase a Catholic is a subject complement. A subject complement serves to assign a property to the subject (or sometimes to describe identity, but not here). The verbal phrase be raised can take a simple subject complement, just like other passive forms of verbs that normally take an object and an object complement:
I considered her a Catholic → she was considered a Catholic (by me).
The odd thing about be raised is that the active form does not always take an object and an object complement, especially not in informal speech; and so the passive form with a subject complement is also less common.
I raised her a Catholic → she was raised a Catholic (by me).
2. Using as a Catholic is more common in modern and informal language. There is no difference in meaning. It functions as an adverbial phrase of manner or circumstance. You could say functions as a preposition here; but, more traditionally, I believe is interpreted as a subordinating conjunction with an elliptical clause:
I raised her as a Catholic. → She was raised as a Catholic [is raised].
3. In to be a Catholic, the infinitive to be functions as an adverbial complement of purpose to the verbal phrase was raised. There is a subtle difference in meaning, in that it focuses on the result of the raising rather than its manner, but this is often of little consequence. It is a common construction, similar to other verbs that can take infinitives of purpose:
I constructed this building to be a church. → This building was constructed to be a church.
I instructed the teacher to be strict. → The teacher was instructed to be strict.
I raised her to be a Catholic. → She was raised to be a Catholic.