cida,
Brown stains are likely rust cause by "iron" pipe in your building.
Forget the products a lot of people need needs to remove lime deposits caused by hard water because NYC water is very soft and has very little mineral content.
The only product that will remove rust is weak Hydrofluoric Acid and this is probably the most dangerous of products. I have an ancient bottle of the stuff sold as
WHINK.
It will work on a toilet and will also work on fabric with rust stains. I would not use it on a cast iron tub or sink without testing a spot first but a ceramic toilet is probably fine, and in fact the bottle recommends it.
Whether it still exists and whether it still contains HF I don't know because my bottle is about 25 years old.
If you find it, wear gloves and safety and apply CAREFULLY...an HF burn is like no other. VERY little of the product is needed and there is no need for scrubbing.
My trick is to cover the stain with a thin cloth (like sheet material) and then dribble the WHINK on and let it sit until the cloth dries out. If you just squrt it on, it runs off too quickly. Peek under the cloth every couple minutes
Remember, HF can even etch GLASS...so be very careful.
edit: I checked and WHINK still exists. It seems they even expanded their line of products.
https://www.google.com/search?q=WHIN...dress&tbm=shop