Strictly speaking, the postulates say nothing about compasses, or even straihhtedges/constructions. Also introducing lengths similarly, involves introducing number which is not a "pure" geometry concept. The third postulate just says that a "circle" exists defined by a point and a radii (which also, not a "pure" geometry construct since it involves a metric - i.e. number.

I would say yes, alot of the fundamental proofs while not striclty "incorrect" or false, are rather informal and contain some hidden axioms/circularities.

Tarski put geometry on a more secure footing using first-order logic.

Similar to how Calculus wasn't on a solid logical foundation until Riemann.