[3] (from OP) directly contradicts this, though I am also very confused why real numbers are somehow unorderable, while the integers are not. https://jaydaigle.net/blog/what-is-the-axiom-of-choice/#fn:3
So you can have a set like "all real numbers greater than 0". It doesn't have an obvious first element, even though you can always compare any two numbers and say "this one is less than that one".
There may be a non-obvious first element, i.e. some strategy for picking a first element that applies to all sets. That's the Well-Ordering Principle. Which feels false to most mathematicians, but it's equivalent to the Axiom of Choice, which feels true.
They are orderable ... For all x, y in R, either x >= y or x < y.
But R is not bounded (no inf, sup) and therefore has no minimum and no maximum.
Relatedly, every set gets a supremum and an infumum. (The first two must be members of the set. The second two can lie outside the set).
Without the extension, you need to make an exception for unbounded sets with the above rules.