Dermpath fellowships contain cross-training (or, rather, exposure to) the "other" specialty, i.e. Derm for Paths and Path for Derms. However, that in no way makes a Pathologist a Dermatologist, or vice versa. (Dermpath is a one-year fellowship, sometimes with an added research year).
It's true that some Dermpaths with a Derm background elects to maintain some clinical exposure. However, as Dermpath is generally more lucrative than clinical Derm, it's relatively rare. However, Dermpaths from both backgrounds does sometimes see patients, especially in some of the more perplexing cases (either in person (rare) or through photos).
As regards the scope of this subspecialty, it's basically the dx'ing of diseases of the skin and hair, from innocent nevi to highly malignant melanomas, and yes, Dermpath boarding allows a Dermatologist to do the entire path workup and sign-outs him/herself.
Finally, remember, that it's really the insurance co. or healthcare organization/hospital that decides the scope of work a physician can do. If you have a valid medical license, you're free to perform everything from dermpath to open-heart surgery, even if you're trained in FP or Psych. But it'll probably be expensive for you to try it.