I'm no convinced the issues with the Airport at all, to be honest.
What the AE does (or should be doing) is just serving up the Internet connection being piped to it from the 2wire.
As I was getting no connection from the AE (even to the 2wire) I've plugged the G4 straight into the 2wire so I can administer it directly.
And I still can't send mail, or visit the Autotrader.
Which makes me think there's some issue here with the 2wire and the AE isn't immediately at fault.

With Option 2:

Your computer should pick up an IP of 10.0.1.x - did it?
It shouldn't matter that you can log into it via the Airport. In bridge mode, the router should be practically invisible since the internet traffic is going through it, not to it.

To access the 2wire again, you may need to plug a computer directly into it and manually give that computer a 172.16.0.50 IP address. Failing that, default it and start again...

Have you turned bridging back off now? Can you post a trace route to Autotrader in this configuration? Also, your IP routing table on the G4? We need to fix this issue before anything else...

:edit: actually I can't traceroute to autotrader so that might not be helpful; too many unpingable nodes along the way. Can you trace to your mail server instead?

The 2wire has bridge network unselected.
Pinging smtp.auth.dial.pipex isn't returning any hits, returns a 'cannot resolve smtpauth.dial.pipex.com: Unknown host' message.

I can't reach it either - it's not a valid top domain.

Can you ping Autotrader.co.uk or 193.243.130.95 ?

Can't even connect to Google now...

Pinging that second address gives me a 100% packet loss.

If you can't connect to Google, there's a bigger problem than just routing. Google is not down. Can you ping 8.8.8.8 ?

Can you confirm:

That the 2wire is connected to the internet. There should probably be a green light that says "Internet" or similar.
The 2wire is set to DHCP and NAT enabled with a LAN IP of 172.16.0.1
That the 2wire has sensible DNS settings such as automatic, OpenDNS or Google.

That your G4 has an IP address of 172.16.0.whatever , gateway and DNS of 172.16.0.1

That the G4 is plugged into the 2wire and can ping it.

Spent 90 minutes on the phone with the ISP.
They're no further forward either, and somewhat logically pointed out that if other devices aren't having a problem, then its probably just something unique to the G4.
Now some moths back, Dad downloaded MacKeeper.

Widely thought of as piece of malware, I had funny feeling when the ads on the talktalk site were for MacKeeper... And then the dude asked what I was seeing on their homepage. Didn't match what he was seeing.

So, I'm nuking the [LIFTED] machine and starting again, while pointing out to Dad that really a single processor 1GHz G4 really owes him nothing and a new machine wouldn't hurt.

UPDATE:
Did a fresh OS install on it, and imported his usual items, documents & settings.
Out of curiosity, I logged in using my admin account and had no problems in loading the internet sites he'd had problems with.
Still can't use the web portal for his email, as he's forgotten his password.
My money is on a corrupt file somewhere for his account.

I'm posting this from my iMac, which is just fine, the wifi's fine (as it was before) so really, this appears to be down to a software problem with his account rather than a network issue.

Well, one thing I've learned in this thread is never to attempt to diagnose a problem involving a Mac or an Airport - they're just too damned weird for me. They should be safely locked in the Mac forum away from us mere mortals

Well cheers for all the effort anyway.
If you're curious to know any further details of WTF is going on with the network here I'm happy to oblige.

I have had MTU issues with 2wire routers in the past meaning some sites would load and others wouldn't.

You can try changing the upstream MTU from 1500 to 1492.

Let us know .. always good to see solutions!

Didge.