Unfortunately, this is the kind of problem I could probably resolve in 15 minutes if I actually had hands on the hardware, but is gonna take forever to figure out by remote control.
All of the sudden what I'm seeing has changed from a home setup to an office environment. That changes some things, so I'll answer accordingly.
But first, I do have to reiterate that a Windows LAN domain (for example "TECSVCS") has nothing to do with Internet domains ("mousetech.com"). It's very unfortunate that the same
word is used, because the functions are nothing alike.
Internet domains have no domain controllers. A
LAN domain controller is involved in validating stations attempting to use LAN services, but an Internet client validates directly with the Internet servers it talks to (not the proxies, but the actual servers).
I actually haven't done much with Internet proxies, but all you
should have to do to get Linux a machine working through a proxy is to point the browser to the proxy server. In Mozilla, this is done by issuing the Edit/Preferences menu command, opening up Advanced/Proxies under the Preferences dialog and aiming at your proxy server.
If you installed Red Hat, the "medium" security feature does allow access through port 80, as I recall. The exact rules for RedHat 8 or 9 should be in /etc/sysconfig/iptables .