Is there such a thing as a permanent academic assistantship position?
Yes, this absolutely exists in my field. Such people are typically called "Research Scientists" or "Research Associates." At my last university, a PhD was only a requirement for Research Scientist III. The level of independence depended on your own experience, and I suppose your PI's desires. That is, I would expect each one to have their own pieces of a project, and be on a few papers a year.

I have not heard of one having teaching responsibilities (even just grading) without holding a PhD, since those are often used for grad student funding. I do know of a few people who had lecturer responsibilities without holding a faculty position technically.