Hey adrian24, no offense, but I have the feeling that you might have quickly glazed over the introduction to Mystery Cult and jump quickly to the juicy bits, aka Virtues, Flaws and cool abilities? Your question are based on the mechanics, not so much on the background.
Those Cults are semi-secrets: magi know they exists, they just don't know who is part of them (btw cults should not be seen as evil cultist conspiracy trying to take over the world/Order, they are not Cuthlhu-like cults). And even some magi might declare their allegiance. It is not like if they were doing something illegal or violating the Code.
a) Not everybody want to be immortal. Especially if it means that you cannot leave a certain place anymore, or need to consume huge quantity of virtus to learn something new. Finally people can consider unnatural to live on forever and against the cycle of the world. There are many ways to achieve immortality, some do not requires secret initiation (one example is given in Hermetic Project). None give true immortality, in the sense of undying, unaging and still able to act in everyother way like a normal human.
b) Nope. It does not. Many breakthrough only give a virtue which enable something that would not be otherwise possible. There is no corralation between a Breakthrough, acquiring virtue and/or skill and the limit it is encroaching or breaking.
c) There is a likelyhood that when openting the Arts, some or all Supernatural abilities are destroy in the process. One Founder lost his magical ability over time (Mercere), no reason is given why. Everything related to the Founders is subject to wide speculation. If you scrutinize Bonisagus achievements and try to apply the Breakthrough rules, and the initiation and the teaching of every magus, your realize it is not feasible by RAW. It gives a great freedom for every SG to decide who or what he was and how the Order was really created. There are no right or wrong answer.
d) You are looking at the problem from the wrong angle. First, decide what discovery you want to make (example: scrying the future), then decide with SG and group if it is a minor, major or Hermetic breakthrough (keep in mind that SG can very well let you go through a path which is a dead end, it is not because a magi decide to work on a discovery that he will break a limit, some will never be broken because they are set in the fundation of the Mythic Europe - typically Divine limit). Then only decide what mechanically would make more sense. If it is about scrying the future, it could be a major virtue, which allows you to use Imaginem guideline to scry the past, getting applied for future scrying. If you look at time travel, scrying, the whole package, then it will definitely be a new form (typically this is one of the limit which cannot be broken).
e) Yes. a)No guidelines in the spell book, and very high magnitude, but it would break no limit b) yes, be mindful that scrying at sodales is a High Crime c) Yes, Creo Mentem (sending image, thoughts to the target). If done both ways, you have telepathy. Otherwise, Creo Mentem with Intellego requisit, but magi will be reluctant to have an Intellego Mentem spell on them.
f) Nope. If it is fully integrated in the Hermetic Theory, then it can be taught within a season to any student/magi with MT, even using just Tractatus (open to debate, but reasonable). If it is not integrated, then the magus will teach a virtue (so probably through an initiation), There is no Inferior/superior breakthrough. It is integrated or not (refer to one of my previous post where I am describing the research cycle).
Parma magica does not need a virtue nor an initiation because it is fully integrated in the magical theory. Everything requiring a hermetic virtue means it is compatible with hermetic magic, yet not fully integrated.
I am going to repeat other people advices, for your first game/campaign, stick to the core book, and if you want more info on every house, look at the 3 House of Hermes book. But do not integrate yet the various factions and subdivision of every house. Especially if everybody around the table is new to Ars. There will be too much to process, and people will get lost in trying to find their ideal mage or optimise their development path, and the GM will drowned in all the variants and subgroups. Get a regular campaign going, and let it cover 30-40 years.
As everybody get familiar with the system, the Order and such, start bringing in little bits you like from one supplement or another, one hedge tradition, a couple of cults, a few house mysteries... Not everybody wants the same thing from a game. I have a player who did not want to bother with Spontaneous magic because too much math and parameters, so he went for with No Spontaneous magic flaw (Weak and Difficult Spontaneous magic). Yet everybody agrees that Spontaneous magic is a great tool from Ars magica. After about 50 seasons, only one of my player is showing interest in a Mystery cult (faerie magic and unique range/target), and none is interested yet in Breakthrough altough I gave them opportunity and insight to look into several. They have already their agenda full with regular stuff. One is looking at a familiar, another at opening a Talisman, a third at an apprentice. And nobody is bored or asking for more.