I am not sure if everything is being answered on here, although there are some very good questions and answers. First, we should back up and define a few things.
Fax Server: Generally, a fax server is software that usually resides on a server or fax machine, but at times in larger companies, is dedicated just for that. A fax server allows documents to be faxed over a LAN either from a PC to an outbound fax or an inbound fax to a PC. It converts the file to fax format so it can be transmitted via a modem. Faxes can be sent via the Internet as well but if not done so via an established 3rd party company such as eXtreme Fax, etc. the receiver must have Internet fax capabilities as well.
Multi-function printers or fax machines or any office product can contain a fax server. Most mid to highend MFPs have a fax server in them. Therefore, the fax server allows incoming faxes to be routed to whatever location has been designated and in whatever format has been designated. The two formats are usually PDF or TIFF. So, an MFP (or better to say an MFP with a fax server) can fax to PDF directly to your desktop.
There are also more expensive MFPs such as the Toshibas, etc., which contain their own hard drive and can be configured to either accept faxes only or accept them and send duplicates to other locations. Faxes can also be sent via an MFP (fax server) and along with being faxed to the detination can also have a copy send to a folder for archiving or printed.
How you get the PDF or TIFF file to AC is up to you. Or whether you decide to use PDF, TIFF or convert either after arrival is up to you.
MFPs and fax servers today can do just about anything you want them to do if you look at the right models (and, of course, want to pay for them).
@Grenville Finally...you sound like the perfect user for F.A.P. Why do you think that keeping the imported items outside of AC makes it leaner? Just wondering.