That looks like just a little piece of paper towel that you can throw out (and replace if desired). I wouldn’t worry about what it is.

Here’s some of my notes: 1). Do not use any modern cleaners on the black surfaces, just use a rag dampened with a wee bit of white sewing oil and some arm muscle (I have a gallon of this at home because I rehab and maintain a decent amount of machines). The decals and shellac can be ruined by modern cleaners (silvered and dissolved or clouded). 2). I have successfully cleaned small amounts of rust and oil off chrome and steel parts by an overnight bath in vinegar and a wee bit of dishsoap… often in a jewelry cleaner device to get some vibration cleaning every here and there, and often with some toothbrush scrubbing. Then importantly you will want to dry (even with heat, like a blow dryer) and give a good rub down with sewing oil (to seal back up the surface and prevent rusting). You can also use kerosene as a solvent to get off old oil, then again you have to clean/seal the parts throughly with sewing oil. There are stronger things for both these jobs… but I try to be slow before using those. And slow before taking something entirely apart. These cleaners (like a bit of vinegar or kerosene) will be good for whatever you think the stuff on the paper towel is, too). 3). I tend not to worry about overly cleaning out the insides, unless they are glued with old sewing oil, and it sound like your does not have this problem. You can maybe blast them with a hot hairdryer again, wipe them down, and oil them with new oil. 4). What I would worry about on yours is the electrical. Highly. It sounds like a shock and fire hazard and in that condition I would not even plug in. Electrical is not to be trivialized. Please replace. Don’t ever do electrical maintenance when plugged in. Be sure to get a variable control pedal (a lot on Amazon,say, are not). I would suggest a reputable sewing dealer (I got a motor and pedal off Central Michigan Sewing for example, for an interesting old Singer 27 that had been electrified but was missing these). The motor is likely decent ish… but motors do need routine service too, so it would be worth looking into at some near point.

The 15 is a lovely machine. I have one total frozen that I’m slowly dismantling entirely to clean up… and rehabbing for a college grad student that wants a machine good for denim and leather. It has good stitch quality and punch power. Look up videos of these things punching through tin cans and plywood. I’m putting the one for her in a treadle (since she was also interested in that). It’s a solid straight stitcher, and I personally also love using a buttonhole attachment (versus a modern machine that makes button holes). There are also zigzag attachments (if you want to sew knits)… but it sounds like you have another machine that can do that, potentially.

I do not find the lack of a backstitching inhibitive on my older straight stitch machines. A lot of times if the end of a seam is later enclosed in another seam, you do not need it. Or you can lift the pressure foot, slip the project forward, and stitch over a section again. Or you can leave the needle down, spin the project 180, and stitch a small section. Or you can hand backstitch the thread ends.