The bottom edge of the windshield gasket on both my cars will slip off when I remove the plastic cowl. Seems like it has never completely come off, but I have to slip it back up into place before placing the cowl back. I would say to remove it and try to get it back to original shape and then press it back up where it belongs. If the windshield had been replaced I would hope that the new sealing adhesive has not been over applied and getting in the way of that rubber gasket.

As to the AC air inlet door I do not recall a cork gasket on mine. I could be wrong, just don't recall seeing one. I could understand a cork gasket getting "stuck" and holding that door closed. Silicone grease on the gasket surface may prevent it from sticking.