Do all wild rats carry disease that humans can contract?

Potentially: Have not actually checked all the rats (particularly the human ones), but most of these are in possession of organisms which may produce human disease. The manner in which you contact the critters will determine whether transmission is feasible.

Only infected ones: If you are in direct contact with rodents or their urine/ feces ,contaminated soil or water are bitten or scratched, dieases may be transmitted like Hantavirus Pulmonary Syndrome, Leptospirosis, Lymphocytic Chorio-meningitis, rat bite fever, Plague, Salmonellosis or Tularemia.