Can an airport without an FAA controlled tower have their own tower and controllers? Can an airport not eligible for an FAA tower and controllers choose to fund their own air traffic control tower? Can that airport require pilots to taxi, approach, land, and take off according to their tower's instructions? 

If so, are there requirements for the controllers in terms of certification, or can they hire anyone they deem suitable for their airport? Do they have to coordinate with the FAA or air traffic control system in some special way?
Yes, in fact there are a number of Non-Federal Control Towers (NFCT) in the United States.  These are NOT the same as Federal Contract Towers (FCT), which are funded by the FAA and operated by contract personnel.

NFCTs, on the other hand, are usually funded by the local airport authority.  Some examples (not authoritative or all inclusive): Mojave (MHV), Laughlin/Bullhead City (IFP), and Merritt-Titusville (COI).

Unfortunately, there isn't a consolidated list of all NFCTs.  Rather, they are tracked by their respective FAA region.  Here are the Southern and Western-Pacific Hub Organization Orders that list their respective NFCTs (not sure if they're current).

Operation of NFCTs is covered by AC 90-93b.