quantum field theory - Proof of Spin-statistics theorem

Is this proof of spin-statistics theorem correct?

http://bolvan.ph.utexas.edu/~vadim/classes/2008f.homeworks/spinstat.pdf

This proof is probably a simplified version of Weinberg's proof. What is the difference?

What is the physical meaning of $J^{+}$ and $J^{-}$ non-hermitian operators?

I'm especially interested in the beginnig of proof of second lemma. How to get this:

$$\begin{eqnarray} F_{AB}(-p^{\mu}) = F_{AB}(p^{\mu})\times (-1)^{2j_{A}^{+}} (-1)^{2j_{B}^{+}} \\ \nonumber H_{AB}(-p^{\mu}) = H_{AB}(p^{\mu})\times (-1)^{2j_{A}^{+}} (-1)^{2j_{B}^{+}} \end{eqnarray}$$

Also why under CPT field transform as

$$\begin{eqnarray} \phi_{A}(x)\rightarrow \phi_{A}^{\dagger}(-x) \times (-1)^{2J_{A}^{-}} \\ \nonumber \phi_{A}^{\dagger}(x) \rightarrow \phi_{A}(-x) \times (-1)^{2J_{A}^{+}} \end{eqnarray}$$ conjugation is from charge reversal, - from space inversion and time reversal. What about $(-1)^{2J_{A}^{-}}$?

Where can I find similar proofs?