When there's any doubt that the care and health of the patient will be negatively impacted in any way (even if the negative impact will be due to the person being flustered by trying to act in an abnormal manner), you err on the side of caution and do whatever is needed to help the patient.
With that disclaimer out of the way, the Rama in Shulchan Aruch 328:12 writes:
כשמחללין שבת על חולה שיש בו סכנה משתדלין שלא לעשות על ידי עכו"ם וקטנים
ונשים אלא על ידי ישראל גדולים ובני דעת:
הגה: ויש אומרים דאם אפשר
לעשות בלא דיחוי ובלא איחור על ידי שינוי עושה על ידי שינוי ואם אפשר
לעשותו על ידי עכו"ם בלא איחור כלל עושין על ידי עכו"ם (אור זרוע ומגיד
משנה בשם ראב"ן) וכן נוהגים אבל במקום דיש לחוש שיתעצל העכו"ם אין לעשות
על ידי עכו"ם (תוספות ור"ן)
The Rama writes that if it's possible to do the actions with a Shinui or through a Non-Jew without having any negative impact (such as a delay), then one should try to do so.
The Rambam also writes that when one has the option of minimizing the number of Melachos involved while saving a life, one should try to do as few as possible (per Shabbos 2:8, with the Ma'aseh Rokeach commenting that it's obvious that we're dealing with a case where both options are equally effective and accessible).
It's worth noting that the Rambam in Shabbos 2:3 writes that one should not do these lifesaving through a non-Jew, child, servant, or woman. Even though it might mean more Issurim are done if an adult male Jew does the actions, due to an additional factor ("lest they consider the Sabbath a light matter") we say the actions being done should be done through an adult Jewish male. This seems to be (based on the simple reading) the opinion of the Mechaber in the Shulchan Aruch as well. (Obviously this is only true if the Jewish male can perform the lifesaving actions as well as the non-Jew.)
In practice, if Chas Veshalom a situation arrises where there is even a slight doubt of Pikuach Nefesh and one is uncertain what to do, one should do everything to help the situation without regard to any Hilchos Shabbos. Afterwards, one can speak with a competent Halachic authority to find out how one should act in a future parallel situation should it Chas Veshalom come up again.
To apply the above to the examples mentioned in the question:
- in Jerusalem, one can call an Arab taxi driver and wait for 5-10 min or use a Jewish friend with a car: If the situation is time sensitive and the 5-10 minutes make a difference (or if there is even a suspicion that it will), use the Jewish driver. If those few minutes don't matter (or the non-Jew is a better driver, knows the way, etc.), one should try to use a non-Jewish driver. However, it would seem according to the Rambam and Mechaber, since we want to avoid using a non-Jew, it would be better to have a Jew drive.
- a Jewish nurse can use a regular stethoscope or a computer scanner: if both are equally effective, a regular stethoscope would likely involve fewer Melachos so that should be used.
- in the emergency, there are two operating doctors: a Jewish professor or an Arab stagiaire: One should be treated by whichever doctor is the 'best' (see Kitzur Shulchan Aruch 192:3).