Does the set $\mathbb{R}$ of real numbers, with its usual ordering, have any axioms, or do all of its properties follow from the construction of real numbers (e.g., Dedekind cuts)?
Some analysis books state the least upper bound property as an axiom. It's not really an axiom since it can be shown after constructing the real numbers. However, there are several other facts which are called axioms in some books, but I am unsure of which are the true axioms. E.g., is the fact that $$a\le b \Rightarrow a+c\le b+c$$ an axiom or not?
This depends on how you define the reals. It's perfectly reasonable to define $\Bbb{R}$ as a complete ordered field, in which case you have the field axioms, the linear ordering, and the l.u.b. property all as axioms.
It's also possible to start at the bottom, with the Peano axioms or whatever, get natural numbers, integers, and rationals, and then construct the reals from there. In that case almost nothing about the reals is axiomatic.
Which is the "right" or "real" way? Neither. Sorry.