Hi, Alice. I'm not sure from your description if it is really peel or being mechanically scraped off, but I know that I would do one of 2 things if I were you. One option would be to chemically strip your moldings. Apply masking tape and heavy plastic wrap or aluminum foil to the walls and floor tp keep from harming them, use a stripper like Citristrip that is relatively safe with a putty knife to remove the stripper/old paint gunk on the flat parts and a stripper pad on the curves. Since it sounds like this paint is relatively fresh, it should be ready to come off in an hour or so. When you have stripped all the old paint off, rinse it off with Citristrip Paint Remover Wash or mineral spirits on a rag. Don't forget your chemical resistant gloves and eye protection and leave the rags outside for a couple of days before you throw them out.
The second option is to sand. I don't like this as much as stripping because stripping does less damage to the wood. But why not use a finish sander that you can buy at a home center for around $25 on the flat parts of your molding. Use 100 or 150 grit paper and don't forget your dust mask. Use a sanding block with the same paper on convex curves and about a 6" long section of dowel on concaves.
On your cabinet--you are headed for a big disaster. You can't stain anything lighter. The poly and stain combinations might be OK for very good painters but trying it once was enough for me. Any runs or sagging show up like crazy. I think you would be a lot better off to strip, try to blend the scratches in with a dark stain and then put on a couple of coats of fresh finish, poly if you like. Lighting/bleaching an old stain is tough.
One other suggestion. Don't hide your questions in another post. Start a new topic so your question gets the attention it deserves.