Are undergrads applying to top math Phd programs expected to have taken core graduate clases? Is doing well in the standard core undergraduate math courses (complex analysis up to the Riemann mapping theorem, abstract algebra including Galois theory, point-set topology, and algebraic topology, as well as integration on real manifolds and Stokes' theorem) good enough to get into a top-20 math PhD program?
I started with linear algebra after placing out of second-semester calculus. I ended up taking almost all the undergraduate courses offered at the liberal arts college I went to but I still think it's not enough, because most of the Harvard PhD students apparently take lots of graduate courses while undergrads.
To be competitive, are applicants expected to have undergraduate research and do additional math programs in other countries (Budapest Semester, for example), and/or take graduate courses while still an undergrad?
In American universities there are more distribution requirements, whereas in universities in Europe, my understanding is that you declare a major right away and only take classes in that subject. Because of that, they learn a lot more math than American students. I've also noticed that at places like Stanford or Harvard, the undergrad senior theses that students write are super advanced and get into current research. Here are some examples: http://abel.harvard.edu/theses/index.html
Are math undergrads expected to know things like Galois cohomology, the Local Langlands Correspondence for tori or Lubin-Tate theory or the Jacquet-Langlands correspondence?
What could I have done as an undergrad to be able to write a senior thesis on stable homotopy theory or the moduli stack of G-bundles? I am thinking that I did not get a good math education in college.
No, possibly except for Princeton.
Berkeley in particular has a long tradition of accepting domestic graduate students who have less background but strong potential.  (I had not learned Galois theory or anything about manifolds when I started graduate school there, and I still have never learned the Lebesgue integral.)
Graduate admissions committees in the US trust what recommendation letters say (with evidence) about the potential of students, and take that into account a great deal, in many cases attaching more importance to this than any actual achievements of students.  Their statistics based on how students have done in their programs and after finishing is that how much math you know coming in actually doesn't make much of a difference; much more important is your demonstrated capacity to learn math and solve problems.
Keep in mind that it is quite possible to do quite impressive undergraduate theses despite not knowing much mathematics if one has a good advisor (who can explain how the problem can be reduced to an elementary(*) one), works hard, and is clever.  (It's quite common for PhD advisors to complain that their weaker students don't actually understand their own dissertations.)  Keep in mind that undergraduate admissions at Harvard or Stanford is extremely selective - generally more so than graduate admissions in fact - and every student at Harvard or Stanford is very clever.
(*) elementary meaning not requiring knowing lots of mathematics, not meaning easy.