In Pedersen Key Distribution, can the public key be persistent?
This question probably goes much deeper than what you actually aimed for. Simply speaking: If you use $(p,g,h)$ just a couple of times, where $x$ is not revealed in any way, then yes you can reuse it. If you reuse it very often you might run into trouble. However in a general ElGamal-like encryption system, you can use a single public key as often as you want.

The tricky part in the security proof is, when you have to deal with security under "selective openings", where you consider a larger number of private/public keys, and grant the attacker the ability to open some of them and want to know if the remaining ones are still secure. A paper about this is e.g. "Encryption Schemes Secure under Selective Opening Attack" (Bellare, Yilek, 2009) or "Possibility and impossibility results for encryption and commitment secure under selective opening" (Bellare, Hofheinz, Yilek, 2009)