Re: Rodgers' identifiable musical style
Pinpointing the specifics of a style is very difficult, particularly for a 'lay listener'. Both you and Dawn have pointed out that, in general, there is/are (at least two) distinct phases in his body of work. This is not only influenced by his collaborators but also by the journey that musical theatre traveled during Rodgers' lifetime - for which, of course, he was partly responsible.
Furthermore, Dawn has raised the point that each of his works post-Hart have distinct stylistic languages that are related to the content of each musical. So to get back to the original question of the thread, I don't think it would be easy for a lay listener to stylistically link two totally contrasting Rodgers pieces, say "To Keep My Love Alive" and "Climb Ev'ry Mountain".
However, using the logic that Yip proposed in his/her latter post, yes, I think it may be possible for a lay listener to recognise certain patterns like upward phrase patterns - like those that exist in "Some Enchanted Evening" and "Getting to Know You". But even that is, I think a little too complicated. You'd have to have a structural knowledge of music, and be aware of how tempo, rhythm and lyrical phrasing can create variations of a structure.
So perhaps a lay listener would be able to recognize a pattern that is point it out. He/she may subsequently be able to identify a pattern and even articulate it, but by then he/she wouldn't be a lay listener anymore.