Does the successor function imply order?

I think underlying this question is the assumption that "order" in PA corresponds to the intuitive notion of "magnitude". In particular, I note that the OP says

I'm not sure that "successor" means "greater than."

This is in fact a very good observation! In general the symbol $ < $ does not necessarily denote a comparison of sizes, and our convention of reading it aloud as "less than" is probably a source of a lot of confusion.

The symbol $<$ means "comes before" (or, if you prefer a single word, "precedes") -- and nothing more than that. $a<b$ means "$a$ comes before $b$". It does not mean that $a$ is "smaller" than $b$ in any sense. There are many ordered structures in which thinking of $<$ as denoting size is really unhelpful and potentially misleading; see Fields that can be ordered in more than one way for examples.

This detail is a source of confusion even in elementary contexts; students often have trouble with the language "$-5$ is less than $-2$", because "less than" connotes a comparison of magnitudes, and all we really mean when we write "$-5<-2$" is that $-5$ is to the left of $-2$ on a number line. It is solely a statement about order, and has nothing to do with size.