Re: You can try already with JCT : Do calculations on encrypted messages
Yup. Unless I missed something, this new release is a better-performing alternative to Gentry & Halevi, using a similar but somewhat different approach. It's all ultimately derived from Gentry's original work at IBM and his PhD dissertation, and he's worked on these algorithms and most of the ones that have been proposed for fully-homomorphic encryption (as far as I'm aware). So the JCT would be a good way to study the ideas; HElib is useful if you want to try putting it into production.
Neat stuff, in any case. The mathematics might be a bit daunting for non-practitioners, but folks with a CS background ought to be able to puzzle out the ideas in general terms.
 Assuming both schemes stand up equally well under attack, which remains to be seen. They rely on the hardness of somewhat different problems, and of course there may always be implementation flaws that expose weaknesses not part of the underlying algorithm.
 Actually multiple approaches, according to the abstract for the BGV paper. You can use RLWE without bootstrapping, RLWE with bootstrapping as an optimization, or LWE. These are variations of the Learning With Error problem, which involves approximating one function from a group of functions, given samples of the function's input and output, some of which are incorrect (noise).