I am following this derivation for the covariant derivative of spinors. I have some questions about this derivation:

1. On page 3 they use the fact, that \begin{align*} V^a(x) = \bar{\Psi}(x)\gamma^a\Psi(x) \end{align*} transforms as a vector. Its not specified what exactly $\bar{\Psi}(x)$ means. Is it the conjugate transpose of $\Psi$? It definitely cant be the complex conjugate of the spinor since it has to be a row vector.
2. I also dont get how they come up with the equality in Equation (19). When I use the parallel transport equation above for the spinor I come up with \begin{align*} S(x+dx)-S(x) = \bar{\Psi}(x)[\Omega_\nu(x)+\bar{\Omega}_\nu(x)]\Psi(x)dx^\nu \end{align*} Where do the gamma matrices rise from?