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If the angle between the vectors $v_i$ and $v_j$ is denoted by $\theta_{ij}$, then using cosine rule we have $$\rho = 2 \min_{i,j \in\{1,2,\dots,m\}}\sin({\theta_{ij}}/{2})$$ Assuming that $m$ is a large number, then minimum will occur for a very small value of $\theta_{ij}$. Hence, $$\rho \approx \min_{i,j \in\{1,2,\dots,m\}} \theta_{ij}$$ Since the points are distributed uniformly, It is not difficult to find the distribution of $\theta_{ij}$ and then find the expectation and variance of $\rho$. This I guess will give a pretty good estimate of the actual value of expectation and variance of $\rho$.