Too late to help now but maybe someone else will benefit. I gel stained my Pella fiberglass entry door 6.5 years ago and it looked great! Used Minwax gel stain and their spar varnish, as directed by Pella. However, it did not stand up to weather at all. It chipped and had to be varnished again every 6 months. This Spring it chipped all the way through to the fiberglass and pulled stain with it! Believe me, I did everything properly. My door faces south east and has no roof. But a neighbor whose fiberglass door faces north had the same trouble - he used the same products as I. My neighbor got disgusted and stripped his down to fiberglass and restained it with Sikkens products 3 years ago. No trouble since. Sikkens makes a Door and Window stain that does not require varnish over top. Fortunately for me, just before I started stripping about 36 layers of varnish off my door, it delaminated! Pella brought me a new unfinished slab (under warranty) and I'm using Sikkens to stain it. I've been told the varnish is the trouble. It's like putting a sheet of glass against the door, trapping the heat. Plus it isn't flexible enough to expand and contract. If I put up a storm door to protect it, the problem would've been worse due to trapped heat. Sikkens is specifically made for fiberglass doors. I don't have any vested interests in the company but thought I'd try to help the next guy... Here is a link that might be useful: Sikkens Door and Window stain

I used a birch-top Canadel table three meals a day for 5 or so years and never cleaned it with anything other than just a dishcloth. We did use place mats, so plates didn't slide around, but I never felt I had to polish out any scratches. When we sold the lake house, the buyers asked for the Canadel set because it matched the bar chairs, and to my sorrow, my DH let it go in negotiations. The day after they moved in, I got a frantic call from the wife -- her DH had sprayed it with Pledge and it looked awful and streaky and she was afraid he'd ruined it! I told her just to wash it with soap and water and eventually they'd get all the pledge off. It must have worked, because I didn't hear back from them. I'm planning to get another Canadel set for our breakfast table in this house.

You find that kind of dark shadowing even on newer furniture. We have an oak living room set, Amish made, and on one arm chair, the dark has started where body oils have accumulated on the wood. It's almost 'gunky' and you feel like you could scrape it off carefully. Oils oxidize and darken, just like zits. LOL. Back when I was a child, wooden floors were the norm and not not the exception and wooden floors and furniture was oiled to keep it looking shiny. It was common for the lesser used part of a floor or stair tred to darken from lack of wear. I think your project may turn out to be less of a problem than you think, and the wood looks like it's in good repair. The steel wool and soap will remove paint spatterings. I use it routinely on my wood floors when I find paint spatters, and that's so common when people use rollers, as opposed to brushes.

About the shellac - wiping with mineral spirits should remove grease and dirt without harming the finish, but alcohol will strip the shellac (if it's a mix of varnish and shellac this may not be true, so spot test). I have shellacked trim with 60 years of gunk on it so I've researched and tested this. Once it's clean you can feather out the damaged area with alcohol and touch up the spot with new shellac. Home Depot sells it, you just need to figure out whether you need the clear kind or the orange kind. if you don't mind stripping the patina off the copper you can try vinegar. Or ketchup, that works for polishing copper pots. If you want to keep the patina don't use anything acidic. Maybe try simple green but spot test and rinse well. Do you have a picture of the problem area of the stove? Off the top of my head I would suggest vinegar, simple green, or windex left to soak in for a while and a toothbrush or no scratch scouring pad.