Brother as far the HEC equivalency is concerned its a common sense that you dont need to verify it. As HEC itself has issued you the certificate after due deliberations and scrutiny. I dont know about such certificates but i do hope that it bears the HEC Stamps and seals. And regarding the Character certificates you can call the BPSC and ask them as you stated that you were a private candidate. Still they have mentioned "Your last attended institution" so whatever be your last attended institution you can call them and ask for your character certificate. Otherwise like i mentioned you can call BPSC and ask them that can you submit three character certificates from three different gazzetted officers. Call and ask them.
Regards