The rock may have been exposed to chemical weathering. Acid rain or being covered in acidic oak leaves, as well as lichens, bacteria, algae and fungi can all affect the chemical surface of a rock. Exposure to dampness can cause migration of some minerals to the surface. Limestone varies in hardness, from 3 to 4 on the MOS scale, and it is porous.

Stains can often be removed by mechanical means (scrub brush) along with ordinary powdered laundry detergent. Mildly acidic mixtures can also be used. HERE is one article on cleaning. You can also buy neutral stone cleaners. Good luck.

I like it as it is. It has a good patina. How do you want it to look?

It is ugly and offensive, there is no way to fix it. If you want to get rid of it will give you my address where you can drop it off in the north east corner of my front yard.

I use Wet and Forget on old tombstones in a cemetery that I help maintain. It takes time but it really does work.