One thing you should do is update your Java. It is way out of date.
Go to Add/Remove and uninstall ALL entries indicating Java.
Then go to Java Software
download and manually install the latest version.

Now as far as your Windows Updates, are you using Internet Explorer to download? They will not download with another browser. You also should turn off all unnecessary programs while doing the downloads, this includes Windows Defender and possibly your firewall and anti-virus program. Try this and see if this works.
What sort of message do you receive when the downloads WON'T install? Do your reboot immediately after the downloads? Some downloads require a reboot before they will install.

I disabled the Messenger service about a week ago and the msmsgs.exe shutdown error still happens.
Messenger is not disabled. It WAS running during your HJT scan as shown by this entry in running processes;
C:\Program Files\Messenger\msmsgs.exe
***************
Additional Info;
I have just checked through ALL available updates for XP ....I do not see any that would apply to your computer; All available updates that I see that would apply to what I see in your log are dated February 13, 2007. You say you did download and install successfully on Feb. 14 so I would have to assume that you do all ready have all updates which apply to your computer.

There ARE updates for IE 7, which you do not have installed, so even if you tried to download they would not install because you do not have Internet Explorer 7 on the computer.
But otherwise I don't see any which would apply to you. All security updates are dated BEFORE Feb. 14 so you should all ready have those.