To get an upper bound (for the first part of the question), one can use use the estimates described in the answers to this question. Then, if the volume of the quotient by the lattice generated by the first $n$ vectors is $x,$ a coupon-collector argument gives an $n + x \log x$ upper bound (one has to be a bit careful, since the expectation is not enough, one needs information about the distribution. For the second question, you want estimates on the singular values of an $m \times n$ matrix, which is a much studied subject, especially lately by Rudelson and Vershynin. You can check out Roman Vershynin's surveys on the subject.

To get an upper bound (for the first part of the question), one can use use the estimates described in the answers to this question. Then, if the volume of the quotient by the lattice generated by the first $n$ vectors is $x,$ a coupon-collector argument gives an $n + x \log x$ upper bound (one has to be a bit careful, since the expectation is not enough, one needs information about the distribution. For the second question, you want estimates on the singular values of an $m \times n$ matrix, which is a much studied subject, especially lately by Rudelson and Vershynin. You can check out Roman Vershynin's surveys on the subject.