Caroline was skiing when she broke her leg.
Is when a conjunction or adverb? Can I replace when with the word and?
Are there ways to rephrase this sentence?
The sentence as written conveys clear information. You could rewrite it as "Caroline broke her leg while skiing" and keep the same meaning.
Both your "when" and my "while" clearly connect the two events (skiing and breaking). If you substitute "and", you name the two events but they are not so clearly connected to each other. "Caroline was skiing and she broke her leg" could simply be naming two unconnected events that happened last Tuesday (for example). To be sure, the speaker probably meant "skiing when she broke" when he or she said "skiing and she broke".
"Caroline broke her leg while skiing" clearly conveys the circumstances surrounding the accident - she was skiing. "Caroline broke her leg and was skiing" is not quite as clear. I would respond to that statement by asking, "Do you mean Caroline was skiing with a broken leg in a cast? Awesome!"
As noted in the comments, "when" or "while" connect clauses, and are therefore conjunctions.