Can regular ortho doctors fix broken fingers just as well, or do I need a hand surgeon?

Yes. Ortho docs are cable (and often interested) in treating hand fractures, most of which can be treated without surgery. However, there are specific hand fracture types that are best treated by an experienced hand surgeon, such as pip joint fracture-dislocations, displaced collateral ligament avulsion fractures and complex fracture-dislocations of the thumb.
Maybe, maybe not. Hand surgeons may have background training in orthopedics, plastic surgery or general surgery, but all have additional expertise and training in surgery of the hand. Hand injuries are common and the best results are generally seen when the first treatment is the best. Most injuries also require specialized postop hand therapy to achieve the best result. See a hand surgeon.