Mission completed! I did enough to be able to abandon this project, and show some after pictures that don’t reveal everything I didn’t do. Success! Oh yeah, and I tried out some more no poo methods which I now can recommend with a clean conscience.
I cleaned the stove according to this recipe: Sprinkle some baking soda onto the plates, add some drops of white vinegar, let it sit while you do the rest of the cleaning. Then you just wipe it off. Before wiping the stuff off, I used one of those metal brushes (I don’t know what the technical term is, they are specifically for these kind of stoves). I found the recipe for this on a Swedish site, they had a lot more useful tips on cleaning, mostly also no poo: Bästa städtipsen för ett fint hem.
For all of the surfaces (cupboard doors, tiles, working spaces, mopping the floor) I used the white vinegar/lemon/water mixture again. For the sink I just cut a lemon in half, rubbed it in and let it sit for a while before rinsing with water. Again, I am really satisfied with the results, so I am definitely done with the poo that we have standing around here.
I found another link with no poo cleaning methods on ByzantineFlowers (I really find this blog inspiring, makes me wnat to try out a bunch of their recipes!): DIY Citrus Cleaner – you’ll find a lot more no poo alternatives than what the title reveals.