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 (where the above paragraph is plagarized from) and also here for the link to the Martin & Wong paper.
