Not if Complex. Although many regular doctors can treat a puncture wound if infected or complex such as underlying muscle injury then see a plastic, ENT or oral surgeon.
Your regular doctor. Most of these wounds can be taken care of by your doctor, unless they are complicated, then you can go to er, any er doctor can take care of a pucture wound of lip.
Probably. If he has had the right training he can take care of it. If not he will send you to someone that can.
Depends. Depending on the size and location of the wound, you may want to see an oral surgeon. An oral surgeon specializes in wounds to the mouth, lips and facial area.
Most likely he can. Your family doctor could handle it or if he could not he can recommend a specialist for you.