Based on my limited experience with catapults, I'll try to answer:
1) Yes, I think you can have a Siege Operator build/destroy the catapult to increase skill. You didn't mention the Siege Workshop a 5x5 structure, but I assume you know that is where the components are manufactured. I hadn't thought of upskilling the engineer through assembly/disassembly, but that would certainly save more wood than my current approach of creating lots of catapult parts. As for which results in faster training, I dunno.
2&3) When loading, the Operator stands in the middle of the device. It appears that the remain there until it fires, but both of my Operators are far enough up in skill level now that the Fire Catapult command seems to happen while the Operator is jumping off to reload. Based on my layout, the Operators are free to walk through/across all 3 tiles aligned along the firing axis.
I also seem to remember other forum postings where people showed off their front gate defenses with ballistae tucked into 3x3 rooms. But the designs were presented in terms of "hey look what I *just* built, it *ought* to crush those gobbers, huh?". (I added the emphasis to point out that the design was untested, although the responders only mentioned that the Operators would likely run away before gettting off a couple shots because they're civilians.)
I hope that helps.