Hi Jim,
Its an interesting problem you have there, but its good to know that your computer has been checked by Marius and found to be clear of malware.
1 You seem to have covered all the usual things, but errors of this sort can sometimes be associated with an incorrect time stamp or incorrect time zone. I glanced at some of your logs and think I remember seeing that your time stamp is correct… however as a first step please double click on the clock in the bottom right of your screen, check all the settings, time zone etc and if incorrect alter accordingly.
If still no joy then:
2 Try re-setting Internet Explorer …..(and use IE for the time being…. make it the default browser) Tools>Internet Options>Advanced>Reset (make sure all security and privacy tabs are set to "default"
You will need to close down Internet Explorer wait 30 seconds and then reopen it to ensure the changes have been "entrenched."
3 If still no joy then try adding your banking website to the "Trusted Sites" on Internet Explorer.
4 Please be sure that all Windows updates are installed (this is really important as many changes have been made to security settings by Microsoft that could affect the errors/warnings you have been receiving) check by running Windows Update …..install any that are shown as available and reboot and then check for any more new updates….(you may have to do this a number of times to ensure that all Windows Updates are installed) Keep updating and rebooting until you get two consecutive notices (after rebooting) that "no further updates are available"
5 The issue of performing a repair install of Windows (preserving data etc) is also interesting!
Microsoft Windows XP media centre installations discs (there are two of them) are scarcer than hens teeth!(This particular operating system was usually installed at the factory and no seperate discs were issued)
A competent local computer technician will have a set but may be a lttle reticent on where they came from!…

There is a chance of course that you have a chum who has a set of discs that you could borrow to perform a repair install (you would need to enter your own Product key from the Certificate of Authenticity usually attched to the bottom ort side of your computer.) however its fairly unlikely you will be able to borrow some disc locally…
6 If you have a good disc image made when the machine was running fast and sweet (made with something like Acronis, Ghost, Paragon, or Macrium etc.,) as you should, then maybe now is the time to dust it off…?
7 There is usually a restore Partition of Windows XP media centre editions that will format your drive and reinstall the factory image so the computer will be just as it was when it came out of factory…… you will have a lot of work to do, in reinstalling programmes, service packs and updates etc (approx 8 years worth!)
8 The restore to factory is not to be undertaken lightly….. in the workshop to reinstall and update everything is a full days work so, in the home, allow much more time…
PLEASE NOTE THE PROCESS IS DESTRUCTIVE AND ALL YOUR STUFF WILL BE CLEARED….(a good accurate and 100% up to date back up of everything on your computer that is important to you is vital. Back up everything you don't want to lose, and make sure you have at least one extra copy kept safely off site on removable media.
9 As you know Microsoft is ceasing support for Win XP on 7th April next year and it will then be unsafe to use it on the internet and for banking or other confidential matters without taking extra special precautions….. to all intents and purposes for the ordinary user it would be as well to upgrade/purchase a newer machine/operating system well before this time, in order to arrange the transfer of documents, data files, worksheets, music, photos etc and to become familiar with a new method of working.
Post back if you need any clarification and let us know how you get on.
Regards
paws