MRSA is short for methicillin-resistant staphylococcus aureus (a resistant form of staph aureus bacteria). It can cause an infection of the conjunctiva (or thin, transparent layer of skin overlying the white part of the eye). It is fairly uncommon, but can be picked up in a hospital setting.
As far as the inflammation in the eyelids, it is very common and it is usually not a visibly obvious finding (the eyelids usually don't necessarily appear red and swollen), but the symptoms that the inflammation causes are quite characteristic as your symptoms sound to be.
As far as the rash coming and going, I do not suspect that it is related to the eyes; usually rashes on the abdomen are not related to problems in the eyes. An abdominal rash can sometimes be related to underlying liver or gall bladder disease. Do you a history of hepatitis or heavy alcohol use? Is this rash itchy?