What does it mean to you to work out all of D+F? You could probably go through it in a semester or so if you kept at it and come out with a decent understanding (most people spread it out over a longer period though with other classes in between and I believe this is a better way to learn...
Depends on how much depth you are thinking of. Almost all areas of pure math will require you to know at least the level of topology one would gain from a course based on Munkres at some point (for instance separation axioms, various forms of connectedness, various forms of compactness...
What exactly does rigorous mean to you? Spivak doesn't get as bogged down in certain fundamentals, but so what? The books cover different things and in different ways. Apostol chose to cover some foundations of natural numbers in a more thorough manner, but this does not make it more rigorous...
I liked Basic Mathematics by Serge Lang. Harder than most high school books, but good if you're really trying to understand the math and have been exposed to it before. It's probably not the most effective if the ultimate goal is simply to score well on the GRE, but if your goal is also to...
Well I have never participated in Putnam or any other college-level competition, but since no one else has offered their advice I thought I might as well give you my thoughts.
You will probably be hard-pressed to find a book that deals exclusively with putnam-level single-variable calculus...
I have an International edition of Apostol (I'm assuming it's the same as the publishers are identical) so I thought I would shed some light on the issue of the quality of the international edition of Apostol since some people seem to be worried about it. It's indeed paperback and the cover is...