Yes, you can show such equiconsistency statements. As Gödel proved, for any set of classical axioms, there is a corresponding set of intuitionistic axioms. And

Yes, you can show such equiconsistency statements. As Gödel proved, for any set of classical axioms, there is a corresponding set of intuitionistic axioms. And if the classical axioms are inconsistent, then so is the intuitionistic equivalent. (Given that intuitionistic reasoning is classically valid, an inconsistency in the intuitionistic axioms trivially gives you one in the classical axioms.) So the care that intuitionists take does not lead to any improvement in consistency. However the two approaches lead to very different notions of what it means for something to mathematically exist. Despite the formal correspondences, they lead to very different concepts of mathematics. I'm firmly of the belief that constructivism leads to concepts of existence that better fit the lay public than formalism does.