I'd replace the firing mechanism and go from there. It sounds like the firing pin isn't a) long enough and/or b) needs a new spring.
Yep ... If there's smoke, there should be "bang". If the firing pin is leaving a strike mark on the primer, chances are its not the gun. A light primer strike is not out of the question but not all that common. Take a photo of the shotshell primer after you have pulled the trigger and post that for us to see.If there is a puff of smoke, it sure sounds like an ammo issue. The primer should either set it off, or not do anything. What you describe is pretty rare for a malfunctioning gun, but not for bad ammo.