What is a Strange Loop?

There are three important strange loops that I encountered, after I realized people are contemplating such a term, proposed and extensively discussed by Douglas Hofstadter in his books.
Self-referential system is the essence of a strange loop of consciousness, see Liar, Epimenides, Barber, Russell's and Quine’s paradoxes, Gödel's incompleteness theorems, Miranker & Zuckerman theory about consciousness.
Chicken or the egg dilemma is a strange loop of evolution. It can be resolved by saying that the egg evolved much earlier than the chicken, although, that egg was not the same kind of egg that appears in the dilemma, ie the one that is laid by chickens and the one chickens hatch from. But still, it's a kind of resolution, provided that we will understand one day the exact mechanism of evolution of multicellular organisms.
DNA or enzymes (involved in protein synthesis and folding and in DNA replication) dilemma, and the information flow network between them, is a strange loop of chemical evolution of abiogenesis. It is a stranger loop in the sense that already von Neumann concluded that information contained in DNA that codes for these enzymes should be immutable, since its mutation would probably break the possibility of self-reproduction, plus, Deutsch emphasized basically that some information in the process of protein synthesis and folding may be contributed by the enzymes involved, and it may never actually originate from DNA (see my last essay in which I first oppose, and then agree with that conclusion), and as such it may also be immutable. It means that once that complex and stable process of self-reproduction appeared and got fixed, it was never subject to big changes, and yet we somehow think it was a result of evolution. OK, but how? Can any mathematician or physicist in the world describe precisely how did it happen, what were the stages in that process, what appeared first, DNA, RNA, or proteins, and how was the current interrelation between them established?
That machinery produces other proteins too, ie the ones that are not involved as enzymes in self-reproduction, and it allows for their variation, regardless of how that variation happens exactly, but there is not much evidence that it allows its own variation. Is there? If not, then what Michael Behe says about irreducible complexity of that system is true, however strange that might sound.
So, that is one interesting question, but to me as a layman, it is interesting also how much variation in protein production that results in new kind of proteins determines overall phenotype variation, for example its body plan, software contained in its neural network and such things? I mean things that basically can be rearranged differently by using the same kind of proteins.
And what does it take to start any strange loop? Take for example a strange loop of me writing these essays, notifying people about it, receiving scarce feedback (therefore I'm grateful for that that I received), and finding new source of inspiration, how did it start? In this case, there was my decision to start it, although, I don't understand quite how I decided it, but that is the whole point of its strangeness.