How does weapon power work in FTL

By "combat", they mean man-to-man combat, not ship-to-ship combat. Basically, Mantes make good raiding parties or security guards.

Breach bombs damage systems and crew, and cause cracks, but don't hurt hulls. Alternate bombing the medical bay and the O₂ generator. Put priority on whichever one the crew is trying to repair. Eventually they'll have so little health they won't try to repair things, and you can just wait. If the ship doesn't have a medical bay it's even easier - just bomb the O₂ generator until they stop trying to repair it.

If you're taking this tack, it helps to upgrade your own shields (and maybe have a defense drone) so you can sit and wait them out without having to waste missiles bombing their weapons as well.

Ion weapons can help, but it's hard to get enough of them to punch through 3-4 shields reliably, while you are still getting shot at. If you have them, great, but don't prioritize them.

You are correct about sensors. Don't even bother trying to kill crew without damaging the ship until you have the first sensor upgrade or a slug crew member.