Hi there,
You are asking too many questions at once. So I'll try to give an explanation that should answer most of the questions you asked.
Anti-matter is formed, just like matter, out of thin air, not really but very close to it. To be perfectly clear, photons carry energy. This energy can be transformed to create matter, following Einstein E = mc^2 where m stand for matter. But in the process of creating matter, anti-matter is also created, following many conservation principles related to quantum mechanics. OK anti-matter is produced at the same time as matter.
So as you said, matter and anti-matter is produced in accelarators, in outer space, but also inside your body and mine. That's right, when a photon (it has to have more than 1022keV) interacts with the matter in your body, pair-production (the term for creation of electron and positron) can happen. Therefore, anti-matter is continously created on Earth, and not just in special labs.
Now to make the little story complete, once anti-matter is created inside your body, it will very quickly interact with the matter around it, anhilating back into energy.
Hope this helps you a bit. Cheers