You cannot put besides and apart from instead of as well as in the first sentence as this wouldn't be correct. The other two sentences would benefit from adding an 'also' before cycling. You could say In addition to cycling, she also goes swimming and running. Not only does she go running, she goes cycling and swimming too. Although she often goes running, she also goes cycling and swimming.

You'd need an 'and' in the first sentence, but otherwise they're all fine, and mean pretty well the same thing.

I think it's correct but the meaning is not exactly the same. I would have wrote: In addition to running, she goes cycling and swimming.