There have been several questions previously in this vein, but yours is more general. My present answer is adapted from an answer to a question asking for a "Road Map" to Homotopy Theory. Your question is a bit different, so I'll write some different things. First, we need to define what we mean by "Algebraic Topology" so I'll take the subfields listed on the wikipedia article. I think the books you have suggested for manifolds (smooth, Riemannian, etc) are already sufficient to get a working grasp.
In general, you have a good list of "first level" sources. The background from the linked answer will give you great second level sources (Q1), plus papers (Q3), for homotopy theory:
Here is a question asks for an advanced beginners book (and here is another, that was closed as a duplicate). The consensus seemed to be that it was difficult to find a one-size-fits-all text because people come in with such diverse backgrounds. Peter May's textbook A Concise Course in Algebraic Topology is probably the closest thing we've got. If you like that, then you can also read More concise algebraic topology by May and Ponto. I also recommend Davis and Kirk's Lecture Notes in Algebraic Topology. I think these would be a very reasonable place for a beginning grad student to start (assuming they'd already studied Allen Hatcher's book or something equivalent). I'll add that nice books for simplicial things include Curtis, and Goerss-Jardine.
Another question asked for textbooks bridging the gap and got similar answers. Finally, there was a more specific question about a modern source for spectra and this has a host of useful answers. Again, Peter May and coauthors have written quite a bit on the subject, notably EKMM for S-modules, Mandell-May for Orthogonal Spectra, and MMSS for diagram spectra in general. Another great reference is Hovey-Shipley-Smith Symmetric Spectra. On the more modern side, there's Stefan Schwede's Symmetric Spectra Book Project. All these references contain phrasing in terms of model categories, which seem indispensible to modern homotopy theory. Good references are Hovey's book and Hirschhorn's book.
Now for Q1 and Q3 for knot theory. A book that was standard reading for graduate students at Wesleyan learning knot theory is here. If the start is a bit rough, I can highly recommend this book for an exposition aimed at undergraduates. After those books, Here is a question that gave recommendations for papers to learn Khovanov homology.
As for Q2, if you take the knot theory route, you'll need to know about the Alexander Polynomial and the Jones Polynomial. A good book to start with would be Dummit and Foote. For any of the above subfields of algebraic topology, it would be good to know some commutative algebra, e.g. Matsumura or (the classic, but harder) Atiyah-MacDonald. For both fields, you also need homological algebra, and a great book would be Hilton-Stammbach. A second level book, more suited for homotopy theory, would be Weibel.
As for algebraic geometry, I have not seen much used in knot theory. If you go the homotopy theory route, you will need to know about sheaves, and eventually about schemes and stacks. A reasonable book would be Hartshorne (but only after the algebraic background above). In my opinion there's no reason to rush into trying to teach yourself algebraic geometry, but if you can take classes on it, do. Much of the algebraic geometry needed for homotopy theory has been reformulated beautifully by Jacob Lurie over the last 10 years, and his writings are also great if you intend to do algebraic topology (after learning homological algebra and learning about simplicial things from the first block of links above).
Lastly, there is the relatively new field of topological data analysis, and applied algebraic topology. Many excellent sources to learn in that field (from the basics all the way up, including what's needed from homological algebra) are at Peter Bubenik's page. In particular, I highly recommend Ghrist's book, and the surveys Bubenik links to.