What sport? Most bony mallet fingers are treated with an extension splint that is worn full time for 6 weeks. If you are playing a sport requiring you to catch a ball, there is risk that you will reinjure the finger and displace the fracture. Once there is evidence on xray of sufficient healing, you would be able to return to sport with minimal risk.
Mallet Finger. It needs to be kept splint in a bit of hyper extension for around 6 weeks. Any flexion ruins whatever healing you get. I would strongly think about how important it is to compete right now and what alternative exercise you can do which won't risk reinjury to the finger.
Yes, most likely. Most soft tissue mallet finger injuries heal well without surgery and with a good outcome as long as the patient strictly adheres to the treatment recommendations. Discuss these recommendations with your treating orthopedic doctor.