(i) The tricky thing about indefinite integrals is that you could end up with different answers with different methods, but all are correct since a hidden constant might prop up somewhere. As long as your method is good, you're good. In fact, if the solution to an indefinite integral is x^2+C, you could have very well written down x^2+10000000+C just to annoy your teacher - technically correct, but why though.
I think there is a classic example of integrating sin(2x) using very different methods, and it will yield different results, but in essence are the same, since they all differ by some constants.
Sidenote: Definite integrals do poses some problems, but at the end of the day it still doesn't matter which path you choose as long as your bounds of integration are adjusted appropriately.
(ii) Another thing with learning so many methods and techniques is not a problem about calculus itself. For all we know, we could only read off the standard result of integrals from a table (or commit to your memory). All these fancy techniques like IBP, u-sub, trig sub and partial fractions is to make something we can't read the result off of the integral table to something we can. As to how we get there, it doesn't matter. Though I should mention, as previous posters would agree, some "shapes" of the integrand do require some specific techniques, which is why practice is so important.
Many people would say, and I concur, "calculus is easy, the hard part is the algebra".