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For me, the definition of $Q$-Fano $3$-fold is a normal $Q$-factorial projective $3$-fold with terminal singularities and $-K_X$ ample. If it is in addition Gorenstein, then this is true since it is locally complete intersection.

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No. He proved that log $Q$-Fano varieties are rationally connected. It is possible that all the rational curves pass through some singular points. I want to know if there are rational curves in the smooth locus. The only result I know of is the theorem of Keel-Mckernan which says that for log Del Pezzo surfaces, the smooth locus is rationally connected.

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Doesn't this paper give a positive answer to your question? (And some more I think)

Qi Zhang, Rational connectedness of log $Q$-Fano varieties, J. Reine Angew. Math. 590 (2006)

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