Can you contract food borne illness by handling raw meat w/ a small open cut or will the raw meat just cause infection of the wound itself?

Infection can spread. Any open wound can become infected from outside contaminants. From there and infection can spread to other areas of the body.
Unlikely. You will not likely get food borne illnesses without ingesting the toxin or organism producing it. Raw meat may have bacteria in it, but they are unlikely to produce infection in a shallow open wound. Wash thoroughly with soap and water.