As to the toilet running now and then - you probably need to replace the flapper. Usually, something (like the flapper) is allowing the water in the tank to leak into the bowl. Eventually, the water level gets low enough that the fill valve lets more water into the tank. This is when it's "running". (If not the flapper, then next is the flush valve.) A new flapper will cost you around 3 dollars and is easy to replace.
I can't explain why dirt forms "around the toilet". If you have dirt/mud buildup inside the tank, that's another matter. I had that problem with my well water when my house was built until I got a filter.

Thank you very much!