Are there a new prime counting functions?

That depends on what exactly you mean by "non-traditional methologies". If you've been here long enough, you might have noticed that a few of my own posts here and on MSE consist essentially of a single computer program (usually in Asymptote) that produces some reasonable approximation to an optimizer or carries out some task within a reasonable time. Often I do not have a formal proof that it works as declared and sometimes I never produce it even later either because I still cannot do it, or because the OP loses interest in further communication. So, when there is a clear problem at hand, especially some applied one, and the standard solvers fail or give clearly suboptimal results, such contraptions may be of interest (to the OP, at least, which already justifies posting them, IMHO).

The distribution of primes has been studied so extensively both analytically and numerically, that to make a contribution that surpasses all that has been done already in at least one respect (strength/clarity/computation speed/range/...), which is the only thing that justifies posting one's thoughts in the public domain, researcher, or not, independent, or not; MO, MSE, a journal, arXiv, or whatever, is rather difficult.

"Rather difficult" does not necessarily mean impossible, so if you really can claim something of that kind, that may arouse substantial interest, whether it is posted within the site guidelines or in a clear violation of them. However, any vague descriptions, unsubstantiated claims, or faulty proofs will quickly move you to the "crank" category in the public eyes, and once there, the way out is extremely hard. So, rephrasing an old proverb, "check thrice, post once".

As to the implied math question, yeah, you can play with primes in many ways. I toyed once with the idea of creating the simplest Boolean circuit that, given the digits of $n$ in binary would output $\pi(n)$ (or $p_n$) in binary for $n<2^m$ (the gates are NOT, AND, OR or, if you prefer to march along with the modern electronics band, NOT, NAND, and NOR, $m$ imput wires, $m$ output wires (for $\pi(n)$), otherwise no hands barred) or, rather, writing a computer program that would design such a circuit (or, if you want to be in the trendy business, train a neural network) in a reasonably efficient way. I haven't had given it an honest try yet, so I cannot tell whether this project might be worth anything, but I cannot tell it about any other project of mine that I haven't completed to my satisfaction yet either. So, yeah, play with everything like crazy (or you'll never find anything interesting anywhere), but apply the strictest sanity criteria when reporting your games in public and get used to the idea that the most often and heavily used object in an office of a mathematician (again, does not matter professional, or not) is the wastebasket.

Otherwise welcome to MO, and let me give you one simple common sense advice that could be applied to any site, group, organization, drunken party, etc., etc. you might want to join: Upon the entry, spend some time listening and observing before speaking and acting yourself.

Edit: I finally found time to finish looking at your opus. My verdict is that you have quite keen powers of observation, but very little (if any) understanding of the basic rules of the game called "rigorous proofs". The collection of things you present is comprised of several curious facts from elementary number theory, a few observations based on the statistical tables coming from looking at primes up to 100,000 or so, some of which are known to humanity as proven theorems and some of which still remain conjectures (alas, most of your "irrefutable proofs" cannot be refuted not because they are impeccable arguments, but rather because they are not arguments at all by the modern (1700AD and later) standards of mathematical rigor). If I understood your logic right, your main formal error is that perfect reflection of all "blades" up to $p_n$ is possible only about the product $P=p_1\dots p_n$, which is huge compared to the right end of the interval $[p_n^2,p_{n+1}^2]$ about which you are trying to derive conclusions from the information about primes up to $p_n^2$.

Your language is quite poetic with plenty of metaphors, but reasonably easy to translate into the standard mathematical formalism when what you are saying makes sense. I certainly enjoyed some of descriptions, though you tend to overuse superlatives.

IMHO, if you invest some time and effort into formal training (supervised, or self-study), you may become a quite successful popularizer of mathematics, should you choose this career path, translating the dry formal proof passages into some vivid imagery without loosing the essence of the matter. However, if you are more interested in the research career, your path to reaching some decent level is going to be rather long. Is it feasible? Yes, except for some really pathological cases, anyone with enough interest and perseverance can learn to do mathematical research. Will it be possible to get there fast? Alas, no, and, as one mathematician once said to one potentate, there are no shortcuts.

Good luck!