1) It absolutely will change. Undergrad math is really not representative at all of what mathematicians do, and you shouldn't get discouraged by it. Math should not be boring and uncreative and pedantic. It sounds like you're not being challenged; take more advanced classes, or start taking seminars and colloquia instead of standard lectures. There's a lot of nontrivial math on the Internet: this board, the arxiv, public lecture notes, etc. For example, Hatcher's Algebraic Topology is available online for free from the author.
2) Although it isn't a very useful one, the most accurate answer for when that point comes is that it varies. I would say that by the second year of grad school--- that is, the point where you've taken the generic required classes and probably finished qualifying exams--- the work should be more interesting than just straightforward symbol manipulation or deduction from hypotheses.
That is, the cutoff point above is classes aimed toward second-year grad students. You don't have to wait until you're actually a second-year grad student to take them.
3) To explain where the lack of novelty is coming from, I think there are several related factors in play.
- There's a difference between doing math and using math. A lot of the undergrad curriculum is designed to give toolboxes to people in other fields, and the proofs offered are desultory and uninspired. (It's a common exercise in calculus classes, for example, to ask students to verify the mean-value theorem by hand for various functions. Why bother?) I think it's a bit like physics: Undergrad Newtonian physics is about using rules like F = ma, conservation of momentum, etc. to solve contrived problems; modern physics is about figuring out what the rules actually are to produce the physics that we've observed. At the undergrad level, you have the definitions given to you. Beyond that, you have to figure out the right definitions that give interesting math.
- There's a much shallower tree of prerequisites for undergrad math, which means that the proofs involved largely follow the definitions, just because there's not much else to work with. If you want to prove something in, say, an introductory real-analysis course, then there aren't many tools available beyond the definitions in the text and some introductory point-set topology. Artin's Algebra has a chapter on Galois theory but only handles the finite, characteristic $0$ case, which means that a lot of the more interesting field theory involved is shoved under the rug.
- Undergrad math is designed for people who are still learning how to be mathematicians (as opposed to the math itself): common proof techniques (e.g., induction), how to construct a proper proof, the difference between rigor and pedantry, etc.
- Undergrad math is old and well-trodden. There are ideas in more advanced math like the Whitney trick that seem incredible at first but become ordinary after using them again and again. The corresponding ideas in less advanced math have already been seen and used again and again, and their novelty has worn off by now.
- Undergrad math focuses on more familiar objects. A lot of the complexity of topology, for example, comes from looking at more pathological spaces. In undergrad math, you really only have vector spaces and some basic manifolds to deal with.
- There actually are some clever proofs at the undergrad level. Check out Proofs from THE BOOK, for example.
