Can non-peer-reviewed work be recognized by the community?
It sounds like there are several questions going on here, so I'm going to try and break them down one at a time.


  As such, my question is, can research be accepted "in any form" by the
  scientific community?


Yes. While the scientific community at large is much more likely to accept research that has been peer-reviewed and published, within a community of researchers things are a bit more loose. Some ideas get passed around on list servers and through emails via pre-prints long before they are published. 


  Does anybody know of cases within the past two decades when such a
  thing (research first published "free on the internet" being
  eventually accepted as mainstream) has happened ?


As noted by zibadawa timmy in the comments to the question, the work of Grigori Perelman appeared on arXiv, was well received and is highly cited. However, as the commenter also pointed out, Perelman is a well established mathematician so their work gets much more notice when something is published to arXiv.


  I just want to know is this even possible in modern science or would
  it just "disappear" in the huge noise of the net.


There is no guarantee that this will happen, nor is this a modern phenomenon due to the internet. Always remember the history of science. Gregor Mendel's "Experiments on Plant Hybridization" is considered to be a seminal paper, but was published in the fairly obscure Proceedings of the Natural History Society of Brünn and it wasn't until after his death that his work was rediscovered and he was properly credited.

To address your work directly based upon the comments to your answer. I would suggest that your best approach would be to get back into university for your PhD so you can receive mentorship and guidance. If that is not possible, your second best bet might be put together a manuscript and send it off for peer review in various journals. If it get bench rejected by the editor then they may provide feedback on better venues for publication. Likewise, reviewers who think the work is worthy of publication in a different venue will also do the same. This could help to find a home for your work.

Also, based upon some of the things you have mentioned with regards to complexity, I would highly recommend looking into the work of Stuart Kauffman and their book "The Origins of Order: Self-Organization and Selection in Evolution" in particular. Some of what you are describing sounds like work that is actively being done and published by scholars that investigate complexity in the context of complex adaptive systems. If that work is indeed similar to what you are doing, then that may point you in the direction of venues for your own publications.