Yes, you get asked about high risk activities while applying for both disability and life insurance. Unless you regularly skydive or free climb, however, I doubt it impacts your rates very much. Check out whitecoatinvester.com and definitely be very well informed about insurance, and/or talk to a financial advisor. As a surgeon you need, at least, specialty-specific insurance (because, if you get general disability insurance, someone could argue that you could still be a radiologist or family medicine doctor). BTW, the emails you get from the AMA about super cheap disability insurance at the end of medical school is general disability insurance. But it is really cheap. A small number of companies even do subspecialty specific. When to start? Depends on your situation. Generally, life insurance is largely to protect a spouse and especially kids. Disability insurance you should definitely get before finishing residency. How soon to get it during residency depends on your risk tolerance. And EVERY surgeon if not every doctor needs disability insurance, not just those who do high risk activities - as mentioned, the reason it probably doesn't effect your rates much is because almost all major injuries happen in daily life... like in a car... rather than some hobby. After 5-8 years of training, if suddenly you couldn't operate, that would be a financial and personal disaster worth protecting against.
About whether your private practice group or co-faculty might mind, I'm not sure. I do recall a surgeon who seemed to get hurt once a year, nothing too serious, but enough to put surgery on the sidelines for a few weeks. At some point, people are going to question someone's judgment and concern for the impacts on others if they keep getting injured.