Easy example of a stable pair in the sense of Hitchin

Question

Let $M$ be a compact Riemann surface with metric $g=hdzd{\overline{z}}$ compatible with the conformal structure. Then

1. The Levi-Civita connection is a $U(1)$ connection defined on the canonical bundle $K$. How to see that?
2. Let $K^{1/2}$ denote a holomorphic lin bundle such that $K^{1/2}\otimes K^{1/2}\cong K$ with the induced $U(1)$ connection. How can we check the existence of $K^{1/2}$?
3. Let P be the principal $SU(2)$ bundle associated to $V=K^{1/2}\oplus K^{-1/2}$. With respect to this decomposition of $V$, define $\Phi\in \Omega^{1,0}(ad(P)\otimes \Bbb C)$ by $\begin{pmatrix} 0 & 0\\ 1 & 0 \end{pmatrix}$. Then the pair $(V,\Phi)$ is stable (in the sense of Hitchin), i.e, for every $\Phi$-invariant rank-$1$ subbundle $L$ of $V$,$\operatorname{deg}L < \operatorname{deg}(\Lambda^2V)$. This is obvious from the definition if we can prove that $K^{-1/2}$ is the only $\Phi$-invariant subbundle. How can we see that?

Thank you for any help or reference in the right direction.