How and when did the argument between creation and evolution begin?

Answer 1

Just after the beginning of the 20th century a religious revival called fundamentalism gained traction in America and the battle commenced. Of course there was opposition before then, but now the opponents of evolutionary science were literalists in their biblical interpretations and well organized in their opposition to the theory of evolution by natural selection.

There is no real argument, Evolution, the change in allele frequency over time in a population of organisms, is fact. The theory of evolution by natural selection explains much of the fact of evolution. Creationism is pure nonsense.