What is the role of a Special Forces Battalion Surgeon?

A battalion surgeon is a doctor, not usually a surgeon, assigned to the unit. It's a position you fill and not a role. No, you will not be able to attend the Q Course, there is no MOS called SF Doctor. You would lose all your certifications in the one to two years it would take to become an SF Medic, Engineer, Commo or Weapons SGT, or SF Officer.
The BN surgeon doesn't do trauma medicine with the 18Ds they teach the 18Ds.

Again, it's only a position you fill and would only fill for 2-3 years till your next job

Greetings. I'm currently in a SF Battalion Surgeon position with 3SFG, so I'll give you my input if it's helpful.

I agree with most of what the others have said and will add the following:

1. Does Guard have SF? Yes, National Guard has SF. 20SFG is in Alabama and 19SFG is in Utah, for example. I don't know as much about them being active component and having few interactions with their personnel, but you can be SF in the Guard.

2. Can you go straight into SF as a battalion surgeon? In the guard, I don't know. But in the active military you can, yes. Typically, the docs that go into SF positions are family medicine and emergency medicine, and they do go right out of residency as their first assignment. One of my colleagues in a sister battalion went right out of FM residency, and one of the senior residents graduating from his FM program in JUN is going to 10SFG. That said, there are others who sometimes make their way in and come from other specialties (like me...as a neurologist). Unlike a conventional army battalion surgeon position that you might get assigned to, however, SF usually has to pick you. I was hand-picked by my predecessor, for example, and had to interview with the battalion commander, Group Surgeon and a few others. They want to make sure you're a good fit. My advice if you're going over as a doctor is to go over and BE their doctor. You're not going on a team, and frankly, they don't need or want you for that. You're there to help them with their health needs, and they have many. These guys tear themselves up and it makes a big difference to know they have a doc who is going to help them get what they need, and is not just trying to get an MFF slot along with a ton of other schools/opportunities. In fact, they generally don't like those folks. But take care of your guys and when you have an 18Z with 20+ years experience tell you they trust you and they've never gotten care they way you have provided it, that's a huge win.

3. Can you attend SFQC and earn the Green Beret? Some docs and many SF PAs have done this, but usually before they went to medical school. Otherwise, that's pretty uncommon. Think about it from a practical standpoint as a clinician. First, you'd have to convince the military to let you not to medicine for approximately year (which is why have you there to begin with) and they you being away from medical practice for a year is generally a bad idea. Your medical knowledge and skills will atrophy quickly. And again, most of the GBs don't want you to be their teammate. They want you to be their doc. If you're good at that, and you earn their trust and respect, you won't need a long tab as a physician.

4. Do you stay CONUS, or can you deploy, go to training, etc? Absolutely, you can. In fact, it's a waste of time to just go to an SF battalion and sit in the aid station. Your primary role will be the medical readiness of the battalion, and as an advisor to the commander on medical issues affecting the battalion. You'll also be responsible for the development of your unit PA, your 18Ds and any other med personnel you have (like 68W W1s). But to do that, go out and do things with them. Go to ranges and pull coverage occasionally. Cover jumps if you're not jumping yourself. Set up lanes for your soldiers. I took a few of my 18Ds to do Grand Rounds presentations with me at the hospital. Go to their AO and see the environments they're working in and help do medical threat assessments. You're not going to be kicking in doors, or finding HUMINT sources, but you can definitely deploy, and should if given the opportunity.

I hope this helpful.

If you attend the SFQC you would become an 18A, no longer serving as a medical practitioner. To become an 18D you would need to enlist as an 18X or a different MOS first then attend SFAS.
MEDCAPs aren't that common for SF any more either, as we have the Civil Affairs branch to conduct Civic Action Programs.
The BN surgeon in a SOF BN provides the license for all of the SOCM qualified medics to operate under, as well as the direct supporting advice and oversee all training, medical readiness, qualifications, and mainly speeds up the process for surgeries and medical procedures due to the Operational needs, maturity level, and speed needed, since the normal Army systems are much slower and based on a majority of lower ranking, immature, Soldiers, that there are many more to cover down on. SOF BNs don't have the manpower to allow someone to remain injured, so each one gets a surgeon/MD.
To enter directly as physician/surgeon you need to contact the Guard/Army Medical Recruiters (AMEDD for Army/Army Reserve), as they deal with all medical professionals and direct commissions, if you wish to enlist for 18X then you need the Guard or Active Army Recruiter - there's no SF in the Reserve but there is Civil Affairs, which also has an Active SOF component.