Why do I get diseased (fever) after I sleep?

It could be because he was attacked by a creature and inflicted, it sometimes take a while to develop, and the game doesn't tell you I think until you take damage.

Also, sometimes if you rest in a "dirty" area you can catch disease, but I am unsure as to where these areas are specifically.

Don't worry, he should usually heal whatever damage is taken while he rests, and I don't think there are any attribute losses. If there are, just rest a week, it should be fine.

How to get rid of disease.
Once each 24 hours - at night I think - the diseased character rolls a Fort check (based on constitution). Two successful checks in a row and the disease is gone.
This means it will take at least two nights rest and could take a lot of time if you are unlucky. This doesn't matter as you are under no time pressure except if you destroy the Golden Orb, so just keep resting until you're healthy.