What is the difference in the application of the article before choosing the side? Or not at all (in what cases)?
Well, for example, such an example: "this is a / the left eye". Why can't we use the article "a" here?
Left is an adjective. It has no impact on the sort of article you want. Are you talking about an eye that wants a definite article, such as the abstract left eye, or when comparing the left to the right (such as on a single person)? Then use the definite article. If it's just any old left eye, use the indefinite.
Adding adjectives to a noun does not affect what article you use (unless it changes which form of the indefinite article is needed, naturally).