I know that institutions must set rigid criteria before accepting research to be conducted by either faculty members or external researchers. My question is: what are they?
Wow, what a broad question. Do you have a sponsored programs office? They should be able to tell you what you need to know.
One thing related to F&A (overhead) is that it can't be arbitrary. Your institution has to negotiate the rates ahead of time for government funding (normally through the Department of Health and Human Services or Office of Naval Research). A good link is here
http://www.research.psu.edu/osp/PSU/Proposal/indirect.htm