Q1. I think, in the medical profession, surgeons (similar to teachers and lawyers and probably some other professions) have to do what is known as a 'residency' or on-the-job training between graduation and being a "proper" surgeon. During this time I believe they shadow experienced surgeons to gain the kind of real world training that university cannot provide... I'm not a doctor though, so it would be good to have someone who is (or who knows with more certainty) confirm this.
Q2. "Within a given month" is a set expression. You will often see "within any given month". It's hard to explain why, but in the sentence "Within a given month, the factory produces one hundred cars" this means *no matter which month* you choose (or which month is "presented" or "given"), the factory produces one hundred cars.
I know that's not the best explanation so I hope someone else can give you a better one..!
Q3. The relative pronoun (in your sentence "which") can be omitted if it is not the subject of the relative clause. And it's neater without the extra word.
I hope that helps at least a little bit!