You want a Pumie scouring stick.
Pumie Toilet Bowl Ring Remover, TBR-6, Grey Pumice Stone with Handle, Removes Unsightly Toilet Rings, Stains from Toilets, Sinks, Tubs, Showers, Pools, Safe for Porcelain, 1 Pack https://smile.amazon.com/dp/B000BPRDCW/ref=cm_sw_r_cp_api_glt_fabc_32M598YBFP2FFV9HP4JP
It’s a magic eraser on steroids for hard surfaces.
Buy two, and the second one will take callouses off your feet better than the pumice stones you get in the Heath and body isle
Pumie Toilet Bowl Ring Remover, TBR-6, Grey Pumice Stone with Handle, Removes Unsightly Toilet Rings, Stains from Toilets, Sinks, Tubs, Showers, Pools, Safe for Porcelain, 1 Pack https://smile.amazon.com/dp/B000BPRDCW/ref=cm_sw_r_apan_i_VG20169D3XHTNK3X09W3?_encoding=UTF8&psc=1
Try this pumice stone for toilets. I've used for strains I couldn't remove and it works great.
Pumie Toilet Bowl Ring Remover, TBR-6, Grey Pumice Stone with Handle, Removes Unsightly Toilet Rings, Stains from Toilets, Sinks, Tubs, Showers, Pools, Safe for Porcelain, 1 Pack https://smile.amazon.com/dp/B000BPRDCW/ref=cm_sw_r_apan_i_VG20169D3XHTNK3X09W3?_encoding=UTF8&psc=1
Pumie Toilet Bowl Ring Remover, TBR-6, Grey Pumice Stone with Handle, Removes Unsightly Toilet Rings, Stains from Toilets, Sinks, Tubs, Showers, Pools, Safe for Porcelain, 1 Pack https://www.amazon.com/dp/B000BPRDCW/ref=cm_sw_r_cp_api_glt_fabc_1FQZ25QHN3EBBGF4ZG98?_encoding=UTF8&psc=1
Pumice stone will remove all the scale and crud and will not harm the finish.
Find one like this: Pumie Toilet Bowl Ring Remover #TBR-6 https://www.amazon.com/dp/B000BPRDCW/ref=cm_sw_r_cp_api_i_cOvVDbZ7SXJF5
So what you are seeing is steel residue, not scratches, you can remove them with one of these.
https://www.amazon.com/Remover-Removes-Unsightly-Toilets-Porcelain/dp/B000BPRDCW
I collect vintage Corningware and nearly the same thing happens from cheap knives being used on the Pyroceram/ceramic, It's actually metal deposits sticking to the ceramic.
These are the best things ever!
Maybe a Pumie?
I used a pumice stone and CLR to remove the ring on my toilet bowl.
I would use a pumice stone for the toilet.
For the tiles, Dawn and vinegar. If that doesn't work, make a baking soda paste and scrub.
I have no ideas for the carpet, though.