Interesting thread. A lot of what I would say has already been mentioned by others, but I'll throw out some thoughts anyway.
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On whether docs would be obsolete:I think there are a number of reasons why doctors would still be around. First, as CollateralDynamo mentioned, people like having people for their doctor. Sure you can program an autodoc to deliver a baby, but what mother wants to drop her baby into the catch-tray of the Obstetricator 2000? We are social creatures and when we are sick or hurt or whatever we need people taking care of us. Plus, a big part of a doctor's job is offering support, reassurance, counseling... essentially placebo- giving the patient a mental framework through which to understand and cope with their illness. My bet would be that head-to-head trials of metahuman doctors vs autodocs would show the metahuman docs having more complications due to error, but the autodocs having worse outcomes overall and worse patient satisfaction.
Second, doctors would be needed for research. Sure you can build a robot that perfectly performs a given procedure, but somebody has to conceptualize that procedure first, and develop it and design all the specialized tools, etc. Not to mention that someone has to recognize the need for such a procedure in the first place (which comes from an understanding of the human condition and human suffering). Granted, with the emergence of AI's between 2060 and 2072, there are now digital entities that might be able to replicate some of those functions (Deus comes to mind... more on him later), but AI's are still rare, those with interest in human medicine would be even fewer. The vast majority of medical research would still be performed by metahumans.
Third, doctors fill a variety of legal and bureaucratic roles. They often function as hospital administrators, community leaders, and are sometimes active in local and regional politics. Unless your machines are recognized as legal entities, metahuman doctors would still be preferred in these roles. Of course, in SR you also have to consider the role of doctors within the corporate structure, as pretty much all medical infrastructure in most countries would be corporate.
Fourth, people might not trust machines to do a lot of things. IRL things like the Tuskegee Experiment and Joseph Mengele's Experiments have had a profound effect on how certain ethnic groups view modern medicine, and those things don't just go away overnight. In SR people would be acutely aware of the events that took place in the Renraku Archology, and that could greatly influence how they view the role of machines and AIs in medicine. If your nurse came in and said, "Our resident AI, Dr.01101110 01101111, has an experimental treatment that he'd like to try but we need your consent..." it would probably give you pause.
On who would become a doctor: I think Orcus is probably right. It would be an elitist thing in many parts of the SR world. You aren't likely to see traditionally trained docs coming out of the SINless population (although there was a great thread around here about non-traditionally trained street docs). However, I think most corporate funded schools would identify potential candidates early on through standardized testing, and "encourage" youngsters to join the corp's medical divisions.
On attributes: I'm sure you've heard the saying "Medicine is a science and an art". I think most skills would link to logic in most situations, but I'd argue that the GM could call for intuition in certain situations as well. And I agree with Summerstorm- not all doctors would have 6s in logic. Most would be 4 (just above average) with solid study skills and a good work ethic. Many would certainly be smarter (logic 5+) and some would be true geniuses (7s) but occasionally you'd have someone who clearly got in because of their connections (3).
On active skills: Axl hit the nail on the head. The skills presented in SR don't accurately represent what a doctor does, but then, its not a RP game about being a doctor (which would be pretty boring IMHO). You'd really need a few more skills. I'd add Nursing to represent actual patient care (of which doctors do very little)- things like starting IVs, administering the drugs, applying dressings, etc. I'd also add Surgery, which is clearly an active skill and should rightly be linked to agility. First Aid is a good approximation of trauma/life support resuscitation (which is based on standardized algorithms, not critical thinking). But Medicine (as its presented in RAW) is overly general. Medicine would encompass things like data collection (physical exam) and basic procedures, so it's still an active skill in some sense, but a lot of what medicine docs do is cerebral- diagnosis, treatment selection etc- and would probably be closer to a knowledge skill. As far as skill ratings, there is a clear hierarchy in medical education that nicely fits the SR skill system.
-- medical student: 2
-- intern: 3
-- senior resident: 4
-- fellow: 4 (with specialization)
-- attending (general): 5 or 6
-- attending (subspecialist): 5 or 6 (with specialization)
-- leader in their field/Nobel prize winner: 7 (this would be famous people like Halstaed, Osler, Murray, etc).
On knowledge skills: most of the basic science stuff you learn in the first 2 years of med school would fit here: anatomy, physiology, pharmacology, microbiology, pathology, immunology, biochemistry, etc. Most docs would also be multilingual, and the good ones would have some street knowledge (especially urban ER docs).
On ware: Senseware would be huge- mag vision, enhance touch, chem sniffers, image link (for laproscopy)- you could really go to town. Sleep regulator is also a great idea (wish I had one right now). Biomonitors (for the pt not the doc). A cyber arm with drug reservoirs and injector. Cyberhands enhanced for surgery (handrazors --> scalpel blades). Honestly, you could do a lot of cool stuff, and being a doctor really gives a PC legitimate reasons to have a lot of non-combat, non-military ware.
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Thats probably more detail than you wanted for your character building "exercise", but if you want to be most accurate you could incorporate some of this and really watch the BP/Karma take off.
Anyway, you should post what you come up with.