Bone is the only tissue in the body that heals itself with a stronger, higher-grade material than the original. So that part of your question has been addressed correctly–the bone itself will be stronger at the former break line than the surrounding bone.
You asked, “Could I ever be a gymnast/martial arts master/stunt man? Or would doctors advize [sic] against it?”…My question is, could you do those things before? Have you a history of disciplined physical training and expertise in maintaining flexibility? The pursuits you mention are very dangerous even for people WITHOUT prior vertebral fracture. You don’t mention how you injured your spine, but adrenaline-charged risk-taking behavior (common among even the most professional stunt people) predisposes one to more injuries and more severe injuries than might otherwise occur. Also, some of the ligamentous tissues (joint capsules, discs) surrounding the area of your fracture likely experienced micro-tears, which heal with a cheap, low-grade fibrotic scar tissue that is prone to re-injury at much lower forces than before. You can compensate to a great degree by building up the surrounding muscles as much as possible.
The idea that “Back pain is typically caused by the discs pressing on nerves that go through small spaces when they exit the spine” is incorrect. Back pain has a list of possible etiologies too numerous to cover here, but by far the vast majority of it is NOT caused by discs pressing on spinal nerve roots. Such pressure is more likely to produce pain or dysfunction in the region of the body serviced by that particular nerve. In fact, many people having a disc bulge or herniation do not even know it until it is discovered by a doctor. Most back pain is typically caused by muscle strain, spinal joint dysfunction (joints moving too freely or not freely enough, essentially), resulting muscle spasm designed to protect the area, and lesions to the facet joints. The facets are the small joints at the back of the vertebrae which are wrapped in ligamentous joint capsules with lubricating fluid inside and are richly innervated with pain nerves. Trauma, slouching, postural abnormalities, etc. can place undue stress on these relatively fragile joints and cause pain and/or an inflammation response.
If you are able to move without pain throughout all normal ranges of motion and have been fully evaluated by a spinal expert such as a chiropractic physician or other qualified personnel, my advice would be to (a) be extra attentive to safety and (b) go for it while you can. In life we tend to regret the things we do not do far more than the things we do.