To remove rust, use a wire brush and then sandpaper. Wipe clean with a moistened cloth. I think the rust comes from acidic food spills, since I get small spots only.
To season, you can use many oils, but use them SPARINGLY. The stove gets very hot and you don't want the excess oil to burn. I did it once and learned the lesson! Better too little and a repeat than a room full of smoke.
There are products called "stove black" or Rust protector like WD40, but their instructions warn not to inhale the fumes, not to spill it on your skin etc. So not exactly a mild natural product 🙄
My grandma's
wood stove never shone although her cast iron pans did. I think it is unrealistic to get that finish on a stove without harsh chemicals. After all, we cook with pots and pans, not by spilling the food directly on the stove 😄