Lurking around MO, I found a question which is related to the second part of my question.  Namely, Greg Martin and Erick B. Wong prove that assuming that the entries of an $n \times n$ matrix are chosen randomly with respect to a uniform distribution from the set $\{-k,  \dots, k\}$, then the probability that the resulting matrix will be singular is $\ll k^{-2 + \epsilon}$.  

See this [MO question](http://mathoverflow.net/questions/90591/singular-matrices-with-integer-entries) (where the above paragraph is plagarized from) and also [here](http://www.math.ubc.ca/~gerg/papers/downloads/AAIMHNIE.pdf) for the link to the Martin & Wong paper.