tl;dr: Zoom is Hunter Zolomon from Earth-2; Jay Garrick is just a fake identity he made up.
The helmet belong to Hunter Zolomon's father, as part of his usual military outfit. When his father went crazy and murdered his mother, Zolomon also snapped, and became a mass murderer. At one point he was capture and committed to a mental ward. While undergoing shock therapy, he was struck by the dark matter and became a speedster.
After escaping from the mental ward, Hunter changed his name to Jay Garrick and adapted the Flash persona. But this was all just a trick: he wanted to give the people of his city hope so he could then take it from them. He explains a lot of this information in the S02E18, "Versus Zoom".
Later, after trying to boost his speed with the Velocity serum, he got sick, and needed to start stealing other speeder's powers to live. This is when he decided to become Zoom. He would act as both hero and villain, to maintain the illusion that the city had a chance to fight Zoom when there really was none.
Once the breaches opened, Zolomon continued to play both sides, Flash and Zoom, but at some point he realized that eventually he would need to be in two places at once without using his speed (since "Jay Garrick" was supposed to have lost his speed). Using some unknown and unexplained process, he went and found a copy of himself in an earlier time that had no speed powers, a "time remnant", and convinced that version of himself to impersonate Jay Garrick. This is the person that was killed by Zoom when they closed the portals.