I wish I had a clear answer for you, I know exactly how tricky it is with having mates forever needing help which means a 70 mile round trip, so I feel for you there.
As to the problem it could be a few things, unfortunately after an infection it could be corruption, but lets hope not.
Now I am a little confused initially, you say you downloaded\installed Firefox ?
You have AVG but yet you can't update it ?
How did you download Firefox ?
My first thought is that you are being blocked by a firewall, AVG does not need to load a web page to update, it simply connects to it's server and downloads any new files, so your browser(s) doesn't sound like it's the issue, it sounds like your being blocked from the net, what firewall is installed ? maybe see if you can check if it has blocked anything.
But you can download AVG updates on another system as a file and transfer them to the other system if needed, although I find that either Avast or Avir are far superior to AVG, maybe consider changing to one of those.
Also windows updates do not need to go to the web site to work, as with AVG it auto checks and downloads any it finds if it's set to auto update, so I don't think any of your browsers is the problem, it's the physical connection itself , so here are a few thoughts, first have you got a system (say a laptop) that you can take and see if the physical connection actually works ? and if you have a spare Ethernet lead try using that for the same reason.
Another thing to check is the device manager, see if any conflicts are showing, go into the control panel and in the admin tools you will see the event log, launch that and see if it is showing any errors, it may give a clue as to what is going on.
Now given the distance here are some more extreme checks to try just incase the above doesn't help, this will try and provoke a reaction that may offer a clue, 1st reboot the router, simply pull the power lead and wait for 30 seconds and plug it back in, does that make a difference ?
Seeing as you have a router there is a fair chance it has a built in firewall, so although this is not a check I suggest easily, for testing purposes for a minute it should be fine, but basically turn off any software firewall, see if that makes a difference (but turn it back on straight after checking), mind you if your using a 3rd party firewall it can still block things when closed, so if it is a 3rd party firewall open it up and check if there are any blocked programs in there.
With the router you will need to check it out online to get the instructions on how to access it, go to the makers site and find the model, and it should tell you a set of digits or similar to type in a browser window, this will gain you access to the router settings, you will also need the user name and password, unless it was changed it should be the default one listed online.
Another extreme is to reset the router, there is probably a reset button somewhere on the back, or near the power jack input, usually a Biro is needed to press it in, again you'll probably find the instructions for this, but on mine it needs to be held in for 30 seconds or similar, this will reset it to the day it was bought, so if it is a deep setting this should clear that.
Now if the physical connection works, and it is just that PC that can't get out then it may be worth trying a repair install, do you have the OS disk ? if so you can try running the sfc /scannow
command, it will replace original system files that may be corrupt\missing etc, failing that boot to the system disk and run a repair install.
As I say these are all extreme options, so only use them if all else fails, but there is an option to make your life easier assuming they can at least get online, get TeamViewer
, install it on your system and theirs, then from your PC wherever you are in the world you can log into the other system and take control as though you were physically on the PC, the only thing you can't do is anything thats required with the system turned off, well offline at least, it means that as long as they can get online with it you can check things out from anywhere, all you need the other person to do is launch the program, and give you the 2 numbers, a 9 digit id and a 4 digit pass, without them giving these codes you (or anyone else) can not access the system, so there is no risk, it has saved me driving 1000's of miles since I started using it, I hope something here helps, good luck.