Sounds as if someone might have dry fired it, poison for a cheap rimfire. Davis is ....okay, perhaps I shouldn't bash, but I've owned a lot of affordable firearms, just don't like 'cheap" ones. My advice is find something better to trust your life to. My North American Arms Black Widow is accurate, 5 shots, and very well made and reliable......if you just like .22 magnum.
That don't help you with your firing pin problem. Try a fix, if it don't work, you ain't lost much in trying. I don't think I'd mess with the firing pin if it's protruding proper, more likely a weak hammer spring even IF the bottom barrel is firing proper. Can the thing fire with several hits on the same spot on the rim of the cartridge? It's not making a deep dent?
Of course, the firing pin could be rounded to the point that it's catching on something like the hole as it protrudes. Push the pin through with a pen or something and see if it's catching on something. In that case, you might try a little light file work. I wouldn't take too much off at a time without trying it, though. The bigger the dent, the less likely you hit an unprimed spot in the rim. Rimfire is notorious for misfires due to inconsistent priming and the bigger the dent, the better.
Gun is too cheap to waste too much money on and a gunsmith would probably laugh at the suggestion, anyway.
