1) So normally, there ISN'T a proxy being used?
Proxies are used in school and corporate environments, and as a mechanism to implement "network nanny" or parental controls, or when information leakage involving sensitive or classified data, or personal or credit card data, or other sensitive data is involved.
When IT folks (or parents) are looking to monitor or control or track network access.
2) A proxy "regulates" what traffic is passed between machines?
A proxy can enforce access, or can cache access, yes.
So no proxy in effect means traffic goes unchecked between computers...?
Correct.
3) Where does a proxy take effect? At the Router/Gateway? The Server? or individual machines?
A functional proxy requires a proxy server, and settings at the firewall, and on the client box.
4) Obviously I have inadvertently set up a proxy somehow over the weekend that won't allow some Safari users to get out of the building.
The proxy setting can be established via WGM, or locally via the system preferences, or (and the following are less common) via malware or such, or due to a bug. These settings are usually altered with WGM and system preferences, in the vast majority of cases.
5) So where the heck did those changes get made?
See 4.
6) In the safari preferences, the change option is greyed out...Yes, I can get to the network proxy area (nothing is checked at this point)...and I see an area in server admin that also addresses the use of a proxy...
If grayed out, then WGM is your next stop.
8) If you have any more immediate thoughts I'd definitely appreciate it!
Without intending any snark, read the manuals? Servers are big and complex and expect an understanding of much more than the client boxes do; that's the inherent difference between "server computing" and "client computing", after all.