I know broken finger has consequences. What are they?

Deformity, arthritis. Most fractures heal. The question is do they heal without deformity? When they involve a joint and are offset(displaced) they can lead to altered motion and arthritis.
Depends. It depends what part of the finger is broken, in what position it is in, and the type of break. If it is managed properly by a doctor, a simple break should have little or no long term issue.