MRSA is spread by contact. So you could get MRSA by touching another person who has it on the skin. Or you could get it by touching objects that have the bacteria on them. MRSA is carried, or "colonized," by about 1% of the population, although most of them aren't infected. That quote is from the WebMD link.
I don't think disinfecting the schools is doing any good at all. Perhaps, they are thinking they have to be seen doing something.
A couple of weeks ago at my daughter's highschool, a girl got a sore on her leg. It was really bothering her and she took off the bandaid to show her friends. A couple of concerned female friends (*not* my daughter, thank goodness!) were touching the wound, remarking that it was warm. It turned out to be MRSA.
Kids are dumb and fearless. They'll come up with things that we can't even imagine, let alone prevent.