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Dear Gil, this is certainly true for random $Q$-acyclic complexes, as you showed in your beautiful paper. But I think for Bernoulli random $d$-complexes of Linial-Meshulam-Wallach, for example, there should only be a relatively small range of $p$ where we see torsion. If it there at all, I expect to only see it when $p = c/n$. Certainly it can't happen when $p \ll 1/n$ or when $p \gg \log n / n$.
updated with an answer to first question, corrected maximum possible genus in terms of number of vertices in second question, updated the third question
