Entering a tradition is entirely a story thing - it's like entering a mystery cult or other society. The exact entry procedures will be different for every society. This could be as simple as having someone to teach you or it could be as complex as a full mystery cult. A tradition is just a group of like minded people who share certain abilities and interact with the divine in similar ways.
For example ascetics are likely to just become a part of that tradition by living as an ascetic - they might become a hermit in the wilderness or join a monastic community like the desert fathers of Egypt. Or they might find another ascetic who can teach them (think of Yoda in Star Wars for that sort of thing).
Cantors on the other hand are part of an actual profession, church hymnists. For them you would have to seek out a renowned Cantor or choir and ask to learn from them, or something to that effect. The teaching of the tradition is combined with the teaching of the mundane skills used by them.
Joining a tradition should be a major plot in and of itself imo, but there is no hard rules as to how it happens.
As to the other points - True Faith is to the divine realm as the Gift is to the magic realm. Favoured abilities for traditions are easier to learn for people with True Faith, but they can also learn any dominion aligned supernatural ability by virtue of having True Faith. A person without true faith is like the ungifted, they have to gain supernatural abilities by initiation and virtues, they cannot learn them by normal means. The advantage for them is without a tradition they would probably not have any way to gain these abilities or to learn how to use them. Having True faith is a big deal, your interpretation in point 2 is correct - the bonus only applies to people with true faith, because the bonus negates a penalty only people with True Faith have.
Characters with True Faith choose a tradition because it makes the traditions abilities favoured for them - this means they can learn them without penalty. Remember - supernatural abilities already known subtract from the xp total for learning new supernatural abilities. A character outside of a tradition has no favoured abilities and will quickly hit a point where they cannot learn any more supernatural abilities because of this penalty. Someone with True Faith joining a tradition is somewhat similar to the opening of the arts for a gifted person. They also still need someone to actual teach them it, and those people are most readily found in traditions (and are less likely to teach someone who is not a member of the tradition). Also, outside of mechanical effects, it is a religious decision and being part of a tradition is pretty significant in terms of character direction - remember, characters with True Faith are touched by the divine and often will want to interact with other people who share that connection (i.e. a tradition).
The reason members of a tradition can learn those abilities mentioned in point 4 is simply because they have access to people who can teach them within the tradition (just like magi can have them because of their education in the order). Taking them at character creation would be fine with me personally, but it's a matter of personal choice for the saga whether that is allowed or if a virtue must be taken for people starting already in a tradition.
People who start with virtues giving them supernatural abilities probably didn't join a tradition to learn them - they joined a tradition because they had those abilities inherently. To gain new supernatural abilities as a mundane you need a mystery initiation. See The Mysteries:Revised for the rules on those, there aren't any examples of divine mystery initiations as far as I know but I would just use the rules from TM:R without alteration.
Hope that helps somewhat.