What are some general guidelines regarding the halakhic permissibility for Jewish doctors to work on Shabbat? I know that saving lives generally overrides Shabbat, but I am curious about less black-and-white situations that may occur on Shabbat, such as having a shift where one must be in the hospital or be on call, receiving medical training, etc. Also, in a practical halakhic sense, are there any situations in which the appropriateness of violating Shabbat would depend on whether the patients are Jewish? I'm sure that these are not simple questions and that there are likely multiple opinions on various issues, but some insight into this topic would be welcome.
I'm sure there are several lectures on YUTorah.org about this; probably an article or two in Journal of Halacha & Contemporary Society as well. But in short:
"having a shift where one must be in the hospital or be on call" -- we generally try to avoid putting ourselves into a situation in which a matter of life-or-death might occur which would necessitating violating Shabbat; this "try to avoid" can be balanced out by other considerations, though. (That's one explanation given for why the Mishna prohibits getting on a ship too close to Shabbat -- a life-or-death situation may occur -- but if taking the trip is a mitzva, it's allowed.)
"receiving medical training" -- a tough issue, as usually the life-or-death situation must be fairly immediate, not "oh this may allow me to save some life out there someplace, ten years from now." Nonetheless, I'm told that Rabbi Dovid Cohen of Brooklyn has allowed people to take residencies that may require working on Shabbat. But check with your own rabbi.
"are there any situations in which the appropriateness of violating Shabbat would depend on whether the patients are Jewish?". That's easy. No. The Halacha is that we violate Shabbat to save any human life; that's the Halacha, that's the practice, that's what we do.
First of all, being a doctor and healing the sick is not considered a Malacha (work on Shabbos). It is considered a Chachma, a gift, not a trade. See Rambam Hichos Refuah.
There is a fantastic Steera (contradiction) found in the Rambam which points this out. In Hilchos Shabbos, the Rambam holds that one is prohibited from carrying an amulet, since (medically speaking) they are of no use. However, in Hilchos Refuah, Rambam states that one can use (carry) an amulet since the ill person believes in this amulet (an amulet must be considered Mumcha, one which has been proven by healing the sick already), and healing is 60% in the mind, therefore the ill will be healed. As you can hear, the Rambam didn't sound like he was in favor of using tricks or the like. But if it works, any port in the storm!
Next question, is one allowed to transgress the Shabbos to heal a Goy? Technically no. However the Chachamim stressed that if they knew this they would kill us, therefore it is in our best interest to heal them ("משום איבה"). I fear to add this but Halacha is just that.
So, can one heal an animal on Shabbos? Basically no. However, one can make the animal comfortable until a doctor can administer to the animal. There is a wonderful book Sefer Tsar Balay Chaim by R' Itchak Nachman Eshkoli written in the last 10 years which deals with anything & everything to do with animal suffering. My son-in-law used this while writing a paper on Feeding Animals on Shabbos while in YU. I highly recommend this sefer to anyone who has an animal, especially since Tzar Balay Chaim is a Torah Issur and it is important to know what one can or shouldn't do.
Medical Halacha is well researched by the likes of Dr. Fred Rosner, Rabbi Dr. Abraham S Abraham (Nishmas Avraham), and many more. There are many Rabbis who write on medical halacha. Although that is a subject for another time indeed!