Book about classical mechanics

I can't believe nobody's mentioned Arnol'd's book "Mathematical Methods for Classical Mechanics" - it covers everything you ask for in the first paragraph quite elegantly (though sometimes somewhat tersely).

Structure and Interpretation of Classical Mechanics (table of contents) certainly deserves mention. It might not have as much differential geometry as you'd like, though they have a followup article titled Functional Differential Geometry.

My favourite for pure classical mechanics is generally the book by Goldstein which includes the Lagrangian and Hamiltonian methods (although I'm not sure about symplectic geometrical and mathematical foundations).

If you want a firm look at curved spacetime manifolds (for vectors, one forms, tangent bundles etc.) I would recommend Carroll's Spacetime and Geometry but it deals with the mathematical backing for General Relativity, which is classical mechanics, but on a curved spacetime.