Even for the case Rademacher entries, Theorem 2.7.1 does not provide a positive answer to your Question 1, which I think does not have a positive answer at all: I think $C$ cannot be made arbitrarily large; or, at best, radically new, so far unknown methods are needed. 

The answer to your Question 2 is positive, in view of Theorem 3.1 and Fact 2.4 (for general subgaussian entries) in the paper by [Litvak et al ][1] (referred to in the book by Tao that you linked). 


  [1]: http://www-personal.umich.edu/~rudelson/papers/rand-matr.pdf