If you haven't been planning for this before you hear the rumor, then it's probably already too late.
1. Circling the wagons is the last resort. You can't win if you're reacting instead of dictating, especially if you're reactions are all defensive. Still, there are times where you have no other choice.
2. Start making new alliances? You should already have 1+ other powers who think you're they're primary ally! You should be working on making friends all the time, not just when you think you're going to get stabbed.
3. Only if it's a good stab. And if it's a good stab, then I was going to do it whether or not there were rumors about me being stabbed.
4/5. I spend a lot of time pointing out to my neighbors how they're much better off working with me than stabbing me. Usually, I'll point out a possibility for a stab, then explain how they have more to gain from working with me than by trying to take my centers, as well as how I can hurt them enough to make the stab a poor choice. Sometimes I'll also bluff by saying I'm going to defend a space or center more vigorously than I am actually planning to. If I'm hearing rumors about being stabbed, there's also a good chance that the rumor is there because the power spreading it stands to gain from breaking my alliance, and I'll often turn that against them by telling my ally about how so-and-so is convinced that our alliance won't last, is planning on us falling apart, and that if we stick together, he'll crumble.
6. I've never really figured out how to use global to my advantage. As far as I can tell, most of the good players pretty much ignore it in full press games anyways.