Remove paint from stained glass windows

Removal is your best bet. Whether you use a heat gun, which much care will be needed since burning what will be bare wood eventually can easily occur. Also you will need to wear a respirator to get past any harmful fumes from the off gassing of the heated paint. Various shapes of scrapers will still be needed to clear out the crevices to freshen up the look of the existing trim. You have some nice trim there it would be great to reveal it from under all the layers.

There is another way top go, with paint remover. There is a brand that uses a cheesecloth that is imbedded into the remover and it allows many layers to lift off at one time. Something to consider and research.

The heat gun treatment will only need a fine sanding and dusting to get a paintable surface. Some paint removers will need a neutralizer in some cases, rinsing in others. All depends on the products recommendations. Fine sanding and dusting is recommended here too. I give you a chance to check the quality of the removal before priming.

Carpets and paints can be made from a variety of materials. Water might clean it up. Paint thinners might dissolve the carpet (depending on the carpet). You could try isopropyl alcohol without much risk. But I would start by testing some carpet fibers (cut a few fibers off with some scissors) in a dish of paint thinner before testing thinner on the carpet. And still, even if the fibers look okay, use caution applying thinner to carpet. Apply thinner to a rag instead, and wipe the paint off with the rag.

If thinner dissoves the carpet, then the last resort is to give the carpet a trim. Hopefully the paint is not too deep and you can carefully cut out just the paint.