Yes.

I don’t think so. If a Jinchuriki dies, I’d imagine the Tailed Beast would simply be free from their human vessels, and maybe find another one. The Nine-Tails, for instance, has had several vessels in the past, and most of them have died by now. The Baryon Mode transformation depleted both Naruto and the Nine-Tails’ chakra, and it wouldn’t stop until either one or both of them had died. Even if the Nine-Tails died, it wouldn’t matter in the long-run, since it’s a being made entirely of Chakra, and it can regenerate after a few years, tho it’d likely find another vessel to be it’s host, such as Himawari.

Yes, that is the reason they tend to defend/protect their hosts. Kurama noted it was going "dark" when Naruto was dying before Tsunade saved him, when Kabuto negated its ability to intervene. Likewise, most assumed that Gaara's sand defence was Ichibi's self-preservation; if they would be freed if their hosts died, no one would assume this.