Success
I now have the remote link working between two PC's with no problems so far, no crashes in Programming Editor or when closing, no invites to send bug reports to Microsoft, downloading, terminal and debug all working fine over the link.
Local PC : Laptop with Windows XP Pro SP3. Programming Editor 5.2.4 installed. Fixed IP address 10.0.0.95
Remote PC : Desktop PC with Windows XP Pro SP2. Programming Editor 5.2.4 installed. VSP.EXE installed. PICAXE connected to COM2. Fixed IP address 10.0.0.10
Network : Wired LAN, 100MHz via switch.
It seems that manually running 'vspdxp_install.exe' solved the problem for me as suggested by Technical in Post #12. I cannot give a detailed 'how to install from scratch on a vanilla PC' as I did a number of different things before getting it to work but this is likely to be the same situation most users are in, so this is how I got it to work, YMMV ...
Start with both PC's off. Boot the PC which has the PICAXE connected to it. Programing Editor 5.2.4 installed, VSP.EXE installed. Run the 'vspdxp_install.exe'. This is a silent install so leave a while, watch the HDD LED activity. Then re-boot.
Run the 'Serial Port Redirector for PICAXE' through the Windows Start Menu -> Programming Editor menu option.
In Step 1, this computer is 10.0.0.10 ( correct ), select COM2 the port PICAXE is connected to.
In Step 2, select 10.0.0.95 ( the other PC ) and COM9, the port which we will later use from within Programming Editor on the other PC to download through.
In Step 3, doesn't work for me because 10.0.0.95 won't respond to a ping. That's an issue which has nothing to do with the redirector so was ignored. It's really just a sanity check.
In Step 4, click "Start Service". I recall some sort of complaint, clicked "Stop Service", then "Start Service" and it worked. If there are complaints about port in use etc, click "Stop Service", change the Step 1 / Step 2 Com ports, then change back to what you want, click "Start Service". Message appears saying "COM2" redirected and not available for other use.
That was it, finished on that PC. My software firewall ( Kerio ) detected and shows SEC_SERVICE.EXE with three ports open; TCP 8888 and 8889 and another it was creating and closing to make a connection out.
On to the other PC, 10.0.0.95, the one which will be running Programing Editor ...
Turned on, ran 'vspdxp_install.exe' on that and re-booted.
Launched Programming Editor, View -> Options -> Serial Port, clicked on COM > TCP.
In Step 1, selected COM9.
In Step 2, selected 10.0.0.10 ( the other PC ), COM 2 ( the port PICAXE is connected to there.
In Step 3, pinging 10.0.0.10 worked, four replies "in 0 milliseconds". Again this is really just a sanity check so actual results may not be too important.
In Step 4, clicked "Redirect this COM Port", list box showed the activity, speaker sounded a "bing bong" during progress ( that hadn't happened previously ) and that was that. If there are complaints about port in use etc, click "Stop Redirecting all ports", change the Step 1 / Step 2 Com ports, then change back to what you want, click "Redirect this COM Port".
When the wizard closed, COM9 became available as a port. Selected COM9, closed the Options window, used the Programming Editor as normal and it works fine with the PICAXE on the other side of the room.
Images - Left, the "remote PC", 10.0.0.10, the one with the PICAXE connected ( Redirector for PICAXE ). Right, 10.0.0.95, the one running Programming Editor ( COM > TCP wizard ).