Cbetting the flop, Fold equity and hand equity etc

[Timlagor]

1. We are IP and we check-back. Our hand equity is 50% (this is our equity vs Villains whole PF range)

That should be our equity against his checking range. Hand equity after he checks doesn't include hands he wouldn't check. Whether that's higher or lower than our equity against his IP range is going to be villain-dependant