Thanks for everyone's attempts at pinpointing the culprit... I'm still getting nowhere, mainly because of the specific blocking of update sites.
Malwarebytes Anti-Malware cannot update, so isn't finding anything.
Copy-and-pasting the link to Microsoft Updates just brings up a 404 server error, which is probably erroneous.
I've run the SmitFraudFix, which has only served to eliminate that as a possibility.
I'm convinced that getting to the bottom of the site blocking will allow me to sort the other problems, but I haven't found anything so far.
There may be a spyware equivalent of a firewall installed... nothing I do with the Windows firewall (including switching it off) makes any difference.
I've already left an HJT log and details of the problem on the TechGuy forum, but that's been there for a few days and nobody has come up with an answer - though there are increasing numbers of similar posts appearing.
I've not had a reply back from Webroot to explain why SpySweeper isn't installing (this problem dates back months) and I'm running out of patience...
The popups are not so much of a nuisance (yet), but whatever is causing them may well be what is interfering with links on websites (I can only get them to work by using copy-and-paste).
Unfortunately I don't have a lot of time to spend sorting this out, so I may well resort to reinstalling this machine from scratch on a new drive on Sunday, leaving the other two until I've more time. The new drive is simply so I don't lose anything I may need and haven't backed up... but will also ensure there are no hidden partitions or boot sector infections.
Alternatively, I've a spare machine I could use for the time being - though I'm concerned at not being able to identify the problem... I've always ben able to sort out my customers' machines, but can't seem to track down the problem on my own system!!