Re: Ars Magica in the 16th Century
It seems to me that the Order would actually have expanded considerably and become explicitly Christian by the 16th century.
This is because of academic magic, one of the more notable hedge traditions. Its methods are very similar to Hermetic magic theory, several Mystery Cults in the Order and among the academics overlap, academics have luck and fate magic, would almost certainly independently develop a Parma Magica in 300 years, and the academics are not penalized by the Dominion.
The Order would be terrifically foolish not to incorporate academic magic into the Hermetic theory. It greatly expands Hermetic power and mitigates the Dominion penalty, meaning that magi could use the Dominion as protection from their greatest enemy, demons, as well as the Islamic sorcerers that would definitely accompany Turkish armies in their attempts to conquer Christendom.
Even if the Order as a whole didn't become overtly Christian this way, some of them certainly would. Even more notably, if such assimilation and merger was avoided despite the many benefits, the Church itself would then develop a massive tradition of powerful magi in the academics, as every university student is technically a member of the clergy. The supply of Gifted children of sufficient intellect to be decent magi would be extremely restricted by competition with the academics, who would be sponsored by the Church.
On other points, I think that guns would be a potent leveler between mundane lords and magi; early guns are unwieldy, but bullets move faster than humans can perceive, meaning that fast casting a defense against bullets would be nearly impossible. Further, cannons would easily level covenant fortifications. When the Order begins to openly counter with magic, the academic magi again emerge, specifically as wizards employed by nobles to protect their soldiers, guns, and powder from Hermetic magical attack.
It bears mentioning that competition for vis would be very intense, which would likely be the primary motivation for the Order to sponsor and encourage expeditions to the New World.
I also think that Hedge Witchcraft would probably be incorporated into Hermetic theory as well, from centuries of exposure if nothing else.