How to simplify a the signing process in an elliptic curve signature scheme that involves a quadratic verification equation?
Is there a way to create a signature passing the above verification process that does not involve a square root in $\mathbb{F}_p$?

Well, one obvious thing to try is setting $R=0$ (the point at infinity); assuming the code doesn't have any protection against that (and the pseudocode doesn't), you compute $h = H_{c_2}(M || R_x || R_y )$ (where $R_x, R_y$ is whatever representation the point-at-infinity has), set $s=0$, and you're done...