Christianity’s rate of growth was actually the same as all other religions. See Not the Impossible Faith, Ch. 18, for data and scholarship.

This is in fact a strong argument against Christianity’s truth: there was nothing at all remarkable about it (see Ch. 2 in The End of Christianity for the Bayesian demonstration of that conclusion).

Paul never mentions killing Christians. He never says in what ways he persecuted Christians, nor even why.

Paul makes clear that Christians were always trying to convert other Jews to their sect. Acts portrays them even entering synagogues, everywhere, to do so. That’s how Jewish authorities discovered them. It would thus be how Paul did.

There is actually no evidence many Jews were even all that concerned with Christianity. Only a rare few ever bothered to actively oppose it in any way. Paul says many Christian churches resided in Judea, indeed even in Jerusalem, still even after twenty years of preaching and recruiting there; so the persecution obviously was fairly mild.

Because the author didn’t know or didn’t consider it important. It hardly matters; good chance no such event ever happened anyway (see On the Historicity of Jesus, pp. 381-833).