Suppose I construct a Pedersen commitment as $g^m h^r$. I could pick the randomness in a "pseudo-random" fashion, such that $r = H(m)$. My questions are as follows:

1. Given that $H$ is collision-resistant, does the security properties of the Pedersen commitment still hold? (i.e. is it still hiding and binding?)
2. Is there a way to construct a Sigma protocol which would prove in Zero-knowledge that a) I know $m$ and b) The commitment randomness is picked such that $r=H(m)$?

Edit: I do not require some specific hash function. Any pseudo-random function would serve my purposes.