The reason we allow someone to violate shabbos to save a life is because of the principle "let him void one shabbat so that many future shabbosos will be kept". What are the parameters for this dispensation? Is one allowed to save an avowed atheist? May one violate shabbos for a non-life threatening situation that may lead to additional sabbath observance (in other words can I break shabbos for kiruv purposes)? Why do we not apply this logic to other mitzvot (let him worship idols one time...)?
Is one allowed to save an avowed atheist?
The Mishna Berura (OC 329:9) writes (my own translation):
A Jew who transgresses averos for his own pleasure, as long as he does not deny the Torah - it seems that we would desecrate Shabbos in order to save him. However if he does so specifically to anger - it is forbidden to save his life even on a weekday, and certainly not on Shabbos.
May one violate shabbos for a non-life threatening situation that may lead to additional sabbath observance (in other words can I break shabbos for kiruv purposes)?
At times we can; Rambam (Mamrim 2:4):
Similarly, if they saw that temporarily it was necessary to nullify a positive commandment or violate a negative commandment in order to bring people at large back to the Jewish faith or to prevent many Jews from transgressing in other matters, they may do what is necessary at that time. To explain by analogy: Just like a doctor may amputate a person's hand or foot so that the person as a whole will live; so, too, at times, the court may rule to temporarily violate some of the commandments so that they will later keep all of them. In this vein, the Sages of the previous generations said: "Desecrate one Sabbath for a person's sake so that he will keep many Sabbaths."
However it seems this dispensation is reserved for Beis Din.
Why do we not apply this logic to other mitzvot (let him worship idols one time...)?
Rabbi Shnuer Zalman of Liadi (author of Shulchan Aruch Harav) explains in Tanya Chapter 24 that the specific averos considered yehareg ve'al ya'avor are a g'zeras hakasuv" (Scriptural decree). He proves this by showing how Shabbos in certain aspects is a more grave offense than sexual offences yet consideration of life overides Shabbos but not sexual prohibtions.