No, the rocks that make up the rings are very small to cross the atmosphere are reach the surface. But, in the case of rings made up by very big rocks, they will only fall arround the equator.
Yes, even the moon or an asteroid may have a ring, in the proper conditions. If you look for "space trash" on Google, you will find an image showing the space trash arround our planet. In some images you can apreciate that the debris created a very very soft ring. Obviously is not a natural ring, but the physics that rule it are the same physics that rule the rings in Saturn.
it would kinda make some observatories useless. it would be a mess trying to look out past it. not to mention satellites and the space station would have to be put in an orbit that doesn't hit the rings. While it may look pretty nice, it would be a pain for anyone dealing with astronomy and etc.