When I studied the equivalents of calc 1 and 2 (from a textbook), the textbook proved everything, although somewhat informally. I mean, with epsilons and deltas and everything, but with a lot more prose than you see in a math paper on arXiv. The different textbook I later used for calc 3, which I actually did take a class in, also proved everything. As did the professor, in class, on the blackboard. I basically never read the book or did any of the homework for that class; I just rederived things from first principles during the exams, based on my memories of the lectures. I always finished the exams last, but I got an A in the class. This was all in the US. My father's calculus textbook, from which I'd learned calculus to start with, was also from his calculus courses in the US.
No in many US universities rigorous calculus proofs are saved for a class called something like Real Analysis which is typically take as the 4th or 5th class for a math major.
Compare e.g.
Transcendental functions, techniques and applications of integration, indeterminate forms, improper integrals, infinite series.
with
Algebraic and topological structure of the real number system; rigorous development of one-variable calculus including continuous, differentiable, and Riemann integrable functions and the Fundamental Theorem of Calculus; uniform convergence of a sequence of functions; contributions of Newton, Leibniz, Cauchy, Riemann, and Weierstrass.
Professors at my university do prove things in the classes I mentioned above, but I was attempting to make a distinction between courses where the emphasis is the proof(Abstract/Contemporary algebra) and courses where the emphasis is the process (Matrix methods). Sorry for any confusion.
I take it my experience was atypical?