Basically, to address the subject matter, "Extended Exasperation Enrages Eternally?" The answer is yes, and that is a good thing. Being self critical of one's work is what in my eye marks the difference between a great artist and a mediocre one. Your outlook, while not universally shared, is normal and healthy among people who wish to build lasting proficiency in a skill set.
Not being completely satisfied with your art means you will always strive to improve. There are many artists who seem to reach a certain skill set, usually around their mid 20's, who become complacent with their ability and never seem to grow any further. In my experience, a good artist is never completely "happy" with his skill set--there is always a higher plateau to reach.
I suspect that while you honestly do believe that you haven't improved over the past 2 years, if we where shown work from year 0 juxtaposed with work from year 2, we would not feel the same. Self criticality does tend to blind your ability to see gradual improvements over a long period of time.
Now, to answer specific questions:
Is it possible that no matter how hard I may try, I'm just not meant to do this? That I'm just not artistically talented in this way?
Is it possible? Yes. However, I think there are very few people out there who are truly hopeless. More likely these people want instant gratification without substantial effort, or they go about learning in an inefficient manner. (There is such a thing as good practice and bad practice.)
Also, "talent" will get you only so far. I was talented at drawing as a young boy, but if I relied on talent alone, to this day I would still draw like a ten year old boy! I suspect about 90% of what I have achieved artistically is thru skill (which is learned) not talent (which is inherent).
Also, where do you get your inspiration to make what you make? Music? Games?
Yes. Also TV, books, friends, walks in the park, head injuries, etc. "Inspiration" comes from a mind that is actively exploring, perceiving, and processing the world about it. As an insomniac, some of my best ideas have come whilst laying in bed with nothing to do but processes the things swimming about my head.
Do you just vomit it out effortlessly?
Sometimes. This is called a fluke. It is exceptionally rare, and it can't be relied on.
And how long have you been at this pixel art hoobie joobie? Am I too young to really do well at this?
I started drawing--in ernest and with a definite agenda--at age 10, which is younger, I believe, that many. Almost exclusively, I focused my skill set on rendering (drawing), and my major duration of improvement was from age 15 to 17. The improvement slowed at 17 due largely to the unbearable inanity of institutionalized education. I didn't get into pixel art until age 18, and the 7 years of drawing definitely provided a strong foundation which smoothed the transition into a different medium. From my experience, high school seems to be the most fruitful period of artistic growth, with college as a close second.