Removing the skimmer plate should not cause the liner to move. I had to replace my faceplate a few years ago. The liner should have gone all the way to the throat so it is fully pinched by the faceplate. Yes, the screws should have been stainless. When I replaced my faceplate I did drain to just below the skimmer opening so I could work on it dry. I pulled the liner back from the galvanized steel pool side and squished a bunch of silicone seal between the liner and the wall, then more under the faceplate before screwing it back in place. I can't see where the liner is torn but you could definitely bond a big piece on as a patch.
Hope this helps.
Al