Can you get rabies if you touch a bike tire shortly after it ran over road kill (raccoon) and you have a paper cut on your finger?

Unlikely: Extremely unlikely. However, if you start feeling ill over the next few days, please consult your doctor. Clean the wound well. Highly unlikely.

highly unlikely: Rabies virus is not transmitted through just a contact. Usually it is through a deep wound, and the virus spread through nervous system. In your case, there was not direct contact to potentially infected animals, and therefore i would say you have no risk of getting infected.

unlikely: Very unlikely. Be careful with the bike, stay safe and consult your dr periodically. Best wishes.