I'm about halfway through a long research project, and still have 5 days left before it's finished. A new project just came up that only takes two days and is a priority project. If I switch to this project will I be able to pick up the old one where I left off?
Yes, you can. You'll pick up where you left off. I'm not entirely sure you keep all of your progress, but a project that was mostly finished will still be mostly finished.
What's more, when you change research projects, if the project required special resources (a corpse or a particular object) then you'll get them back when you change projects, and they will be re-consumed when you resume the project.