Just one question: The HHS mandate is definitely
generally applicable, no questions there. I’ve seen a thread where someone tries to say, essentially, that it does not meet the second criteria you’ve set forth- namely, that it also does NOT refer to a specifically religious practice.
But NOT using any artificial contraception or participating in any form of sterilization or abortion (even by abortifacent pills) is very much
a law of catholic faith!

A consistent** 2,000** year old law that was punished very severely with the most severe penances and ecclesiastical sentences
from the very beginning! So much so that helping in the latter (abortion) results in the automatic application to the soul, of the Church’s MOST severe punishment- Ex-communication, which is exclusion from the means of grace in the church, and from the Catholic pov, is a very bad state of affairs to be in. More so, in Catholic morality (See CCC on morality), giving occasion to others to commit objectively grave immoral acts (like all those three things here), by enabling, helping, advising or even by bad conduct, is scandal- Itself a serious sin.
So, really, why would people conclude that this does NOT concern a specifically religious practice?

For Catholics, it is definitely a matter of religious practice!