Who should do the desecration of Shabbos for the ill person- Jew/Gentile/Man/Woman/Child?
A. First Opinion:
Must desecration of Shabbos be done by an available gentile or child rather than an adult Jew? [Despite the ruling that when the treatment is not urgent, and there is thus no danger involved in taking more time to prepare the treatment in a way that will lessen the amount of transgressions to be done, then one must do so, nevertheless] even [in a scenario that] we have in front of us gentiles and children under the age of Mitzvos we do not say that [since] one is able to [give the treatment] through these people and avoid desecrating Shabbos through a Jew who is above the age of Mitzvos [therefore it should be forbidden for a Jew above the age of Mitzvos to do so. Rather we say that an adult Jew is to do it].
The reason for this allowance is: because since the only option in saving him is through doing an action which is forbidden on Shabbos, therefore Shabbos is overruled on his behalf for [all] Jews which are obligated to save him and there is no transgression of Shabbos here at all.
The Rabbinical prohibition in giving over the job to a gentile or child: Furthermore, even if a Jew wishes to be strict upon himself [to not transgress Shabbos and thus wishes] to do the required labor through a gentile or child, or [he wishes to have them do it] because he does not desire to trouble himself [to do so], there is a Rabbinical prohibition in [him refraining from doing it himself].
The Reason: as perhaps the bystanders will now say that with utter difficulty was [transgressing Shabbos] permitted [to be done] for a life threatening situation and [they will think that] initially it is not permitted to transgress Shabbos through a Jew that is above the age of Mitzvos, and this may then lead to that where a gentile or child cannot be found then they will not wish to desecrate Shabbos through an esteemed religious leader.
May women be entrusted the responsibility of saving lives? [Furthermore] even by women which are obligated in Mitzvos [of Shabbos] and there thus exist no such worry [that if given to them to desecrate Shabbos one may think that doing so is not so suitable] nevertheless we do not place upon them alone the responsibility and duty in dealing [with life threatening situations], that [the treatments] should be done through their hands [alone], as perhaps they will be lazy [in doing so with lack of alacrity] or negligent in it.
Having women take part in the treatment: However women may be joined together with Jewish men, if the man will receive the overall responsibility [of the treatment] and the woman will be directed by the Jewish man, as since the Jewish man is dealing with the treatment she too will do so with alacrity due to his [presence].
Doing all the treatments through Torah Scholars: Nevertheless the greatest form of fulfilling this Mitzvah is to try to have all the [necessary Shabbos desecrations] done through Jews which are great [Torah] scholars and not through the simple folk and women. [Obviously however this should only be done if it will cause no delay in the treatment. Thus one should not search for a Torah Scholar but rather if a Torah scholar is at the scene then he should be the one to do the prohibited actions. ]
The reason for why the treatment should not be done by the simple folk and woman is: in order so that Shabbos not be considered frivolous in their eyes and thus have them come to be lenient even in situations that are not life threatening. As well [another reason that great scholars should do so is] in order to publicly rule through a practical case [that Shabbos is allowed to be desecrated in these situations].
B. Other Opinions:
[However] there are opinions which say that since [in life threatening cases the] Shabbos [prohibitions] are [merely] being overruled and the [holiness of Shabbos has not been] revoked at all, [therefore] whatever [treatment] is possible to be done without Biblically transgressing [Shabbos] is required to be done through not transgressing a Biblically forbidden action and therefore if one is able ………, as well as if one is able to do [the transgression] through a gentile without any delay at all, then one is Biblically required to do so through a gentile.
The reason that a gentile is trusted to do the treatment: [as according to this opinion] it was only [Rabbinically] forbidden to do so [i.e. the treatment] through a gentile in a scenario that there is worry that he may do so lazily and thus come to delay and suspend [the treatment], however [they did not prohibit asking a gentile] if a Jew is actively supervising him and hastening him in a way that there is no worry at all [of possible delay].
C. The Final Ruling:
Halacha like 1st, Custom like 2nd, best to do like 1st: The main Halachic opinion is like the first opinion, although the custom in these countries is like the latter opinion. Nevertheless, it is best not to follow this custom because there is worry that perhaps since people will now see that the treatments are only done through a gentile they will come to think that it is always forbidden to do so through a Jew, and occasionally there will not be a gentile available and the ill person will be endangered as a result of their delay in waiting for the arrival of a gentile.
Those who follow the custom of the second opinion must announce to all: At the very least one who wants [to follow the custom and] do so through a gentile is to make publicly known at that time that it is permitted for a Jew himself to do so [and it is only not being done through a Jew] because a gentile is readily available
Summary-Who should do the desecration of Shabbos for the ill person?
[If there are a number of people to choose from without delay to help save the person’s life then] it should not be administered to women, children, gentiles, or ignoramuses but rather should be done completely by the Torah scholars and leaders. However it is permitted from the letter of the law to allow women to help in the treatment, having a Jewish man supervise. However there is a Rabbinical prohibition to allow a gentile or child to administer the treatment [even if there is a Jew supervising].
Those people of communities which hold of the opinion that argues on the above, and holds that the treatment must even initially be done through a gentile/child under Jewish adult supervision when there is no delay in administering it to them, should not follow this custom, and those who do follow the custom must announce the reason that they themselves are not administering the treatment is because there is no delay involved in having the gentile do it.
[Whenever there is delay involved in administering the responsibility for the treatment to someone else, according to all, it is to be done by whoever has the ability to help immediately.]
Should a gentile doctor be given over the treatment of a Jewish patient instead of a Jewish doctor?
No, specifically the Jewish doctor should be chosen as ruled above.
Should a religious doctor be given to take care of the treatment over a non-religious Jewish doctor?
If they are both equally expertise then the religious doctor must be given the care for the treatment according to all opinions [even according to the second opinion which prefers a gentile over a Jew] being that he will diminish in the Shabbos transgressions when possible.
If however the other doctor is more expertise then obviously he should be chose over the religious doctor.
May a woman nurse be initially entrusted with care for the patient or is one to request specifically a man?
There is no suspicion that these women will do their job lazily and they thus may be entrusted. Nevertheless a man should try to only have a male give him treatment as to avoid the prohibitions involved in touching women.
 The first opinion is the view of the Michaber  in the name of the Rambam.
 Rama, in the name of the Ramban and Oar Zarua.
 So rules Taz.
 So rules also Taz and Mishneh Berurah 37, albeit for the reason that the gentile will do so at a slower pace. However according to Admur, even when we know for certain that a gentile will not do so at a slower pace, nevertheless one is to do so himself.
 So is implied from the fact that Admur only mentions supervision regarding a woman, as well as from the second opinion which only allows it to be done with Jewish supervision thus implying that according to the first opinion, of which we rule in accordance of, this too is not allowed.
 This applies even [and maybe even more so] if the doctor is not religious, as is evident from the reason mention for why a Jew is to be chose over a gentile.
 Piskeiy Teshuvos 328/9
 Ketzos Hashulchan 135 footnote 9