The repair process completed, and reported KB 2538243 successfully installed, thank you!
When I closed the installer window afterward, there was a Microsoft alert window open underneath saying that it might not have installed correctly. This gave me two options, either to state that it had installed properly and to then dismiss that window, or to try the installation again using recommended settings if it hadn't installed properly after all.
So I checked the update log and it showed an entry for the successful installation, so back on the Microsoft alert window I took the first option and confirmed a successful install, and the window closed.
While I was checking the update log for that I had noticed there were 2 important updates waiting to be installed, so I went back to check them and although they were the same two updates - KB 2538243 and KB 2538242, equivalents for C++2008 and 2005 respectively - I am confident they had been just sitting there already since before these latest steps, after being downloaded either overnight or this morning.
I nevertheless decided that I would now click to "install updates" in order to check if they would both now install successfully, and they did in just a few seconds :)
I waited ten minutes just to check that all seemed to be OK, but when I went to close the Updates window which I had overlooked closing before and was covered by my work-window, I saw that another iteration of KB 2538242, the update for C++2005 that had always installed fine, had been downloaded in that short time and was waiting to be installed despite having been successfully installed just minutes earlier and also at least daily for a long time.
So as a further test I clicked "install update" on it right then, checked that it was successful and had cleared, then immediately clicked to check for updates and right away windows downloaded it again and it is now waiting to be installed.
So to summarise, it seems you have fixed both the constant nag-windows for me and also the sticking KB 2538243 update now, and it also appears that there was/is a second *looping* fault as well that just got completely overshadowed ... the constant reinstalling, ignoring, re-downloading, and re-reinstalling of KB 2538242 which relates to C++2005.
If all that that fault does however is to keep going round in circles quietly all to itself, it is liveable-with as long as the underlying installation is indeed going ahead each time perfectly ok and isn't setting some future trap for when I might go to use a programme/utility that needs it, and then can't.
I am happy to follow your guidance on whether that is a last thing that you feel needs to be taken care of to complete everything now, or if you wish to leave it because in reality it is of little real consequence or effect ... I am extremely grateful indeed for ALL your time and attention to deal with the nuisance-effect problems that have now been eliminated.
Just one question either way however ...
... when the repair of KB2538243 went through successfully, the final window before closing advised "It is highly recommended that you download and install the latest service pack and security updates for this product". I have assumed that that is just a standard cautionary notice, however if it is something actually needing to be done, is the "product" it is referring-to the programme/software that C++2008 was installed as part of in the first place, and if so how can I track that down, would there be there an association I could find somewhere in Windows if necessary?
Or is the "product" the C++2008 itself and any continuing updates will therefore already be dealing with service packs and security concerns and I don't need to take any steps?