1. Changing primary and secondary nameservers 2. Moving from one server to another (different ip address) 3. Revamping(?) design and content Will these affect my pagerank? This is for another website, not the link on my signature. Thanks in advance!
If done correctly, changing hosts should not affect your page rank. Now changing design and content will muddy the water. It may not affect page rank but ceretainly can change SERPs. I think the answer here will depend on changes made. Just changing your title can change your SERPs. Good luck with whatever you attempt to do.
None of those have ANYTHING to do with PR. PR is ONLY determined by incoming links to your site. PR = 100% from backlinks. IP (geotargeting) and content can affect SERPS - which is what you should really be worried about.
I dont think it will either, but if themed backlinks have any any more punch than untheamed links, and you change theme in theory you might have same number of links but not on theme of new site and potentially lower page rank.
Themed backlinks carry much more weight with google SERPS, but not with toolbar PR. Indirectly, you might lose backlinks because other sites don't want to link to the site any longer - but if backlinks remain the same, it will not affect PR.