The problem I see with this definition is it doesn't get basic enough. The lone last survivor of Homo ergaster, doomed for sure, is still "life" though not by this definition since she has no capacity to reproduce and adapt. I will assume that by reproduction we are talking in species and by adapt we're not talking about donning a deer hide when it gets cold.
Though AlphOmegskid's definition seems quite good for Earth-based life it does leave out viruses and, most probably, anything non-terrestrial.
The old example of someone on an alien world curious about a rock lying on the surface only to watch it scamper away as he approaches. No details of chemistry or population dynamics but the instant presumption of "life" none the less.
My favorite definition of life is complex chemistry in continuous motion. This leaves plenty of big holes and is not adequate by any means. That vat of fermenting beer in the basement qualifies (and maybe not that far wrong come to think about it) and it takes a stretch to include the dormant stage of a virus, but it is the best definition I have found.
From which you desire to call self replicating RNA molecules as being alive according to your definition. To do this you have to ignore cell theory totally, and there is no good reason to do that.
Is there some reason Cell Theory precludes a self-replicating RNA, subject to copy error, resulting in new capabilities, subject to environmental stress and natural selection, does something in Cell Theory preclude this from a definition of "life"?
Today Cell Theory is a given since the cell is fait accompli and a part of all life. But what about prior? How would Cell Theory negate RNA World or Cairns-Smith or PHA World?
According to Evo theory life began (cellular Life) about 3.5B years ago.
No. Theory of Evolution does not posit how or when life originated. TOE says what happened to that life once it was here and how we ended up with so much variation. That is all. Once we have a viable, evidenced, theory of origin then it will be incorporated our theory of life and evolution. But we don't have one yet. We only have the theory of evolution.