The answer to your question is yes in my experience at major US research universities, assistant and associate professors will, in general, work independently from full professors or individuals with named chairs.
Despite the name, assistant professors are not usually assisting in the research of more senior faculty. There are exceptions including in very large multi-PI labs, in special non-tenured faculty research positions, and in other special circumstances. Of course, nothing keeps junior faculty from collaborating with senior faculty — and many do. If anything though, junior faculty are encouraged to do at work on their own or with their own students to demonstrate their individual intellectual abilities for the purposes of the tenure and promotion processes.
Named chairs usually refers to special permanent funded positions at a university. Usually, these chairs are in a particular area and have been funded to support a particular line of research. As you suggest, named chairs often come with special pots of research funding and are generally more prestigious. Some particularly old or famous named chairs are extremely prestigious.
That said, by no means do all full professors or senior faculties have named chairs. Additionally, it's increasingly common to see junior faculty with named chairs as well including in temporary "Career Development Chairs." Although named chairs are generally more prestigious than non-named positions, they do not usually signal a higher rank (i.e., assistant, associate, full) and they certainly do not suggest a supervisory relationship.