Is it possible that there is a conflict with a previous installation of Morrowind?
Your description sounds like something went wrong with the installation - but don't worry about the installation folder or UAC.
Try uninstalling and reinstalling Morrowind. Ideally you should remove all traces of Morrowind from the registry after the deinstallation - but don't do it yourself if you are not familiar with editing the registry.
Before you do this you could make a final attempt: Don't use the desktop icon to start Morrowind, but go to the installation folder and double click Launch Morrowind there. I don't think it helps, but it can't do any harm to try.
this also happened with Vampire Masquerade Bloodlines CD-ROM installation, no this is the first time I have installed Morrowind but I did it from the GOG Galaxy installer. I will try downloading Morrowind from GOG and doing it manually. as for Vampire Masquerade(maybe a separate issue maybe not) I have attempted to run and install it twice don't know if it is something in that that affected Windows 7 or not but it seems some games, specifically ones made during the time period these two games were made, early 2000's that affects the way WIndows 7 handles them. I have no trouble running newer games nor do I with DOS Box, only games made during 2002-5(I think is when Morrowind and Vampire were made along with a handful of others.)
however, both Oblivion and Morrwind get the "stopped running" error when I exit and that has nothing to do with this issue related to startup except that it might be related to Morrowind's behavior in some way to this. Oblivion, of course was made much later has no problem launching doesn't relate to this except for the exit issue being the same as Morrowind. "stopped running" followed by my keyboard and mouse not working unless I go into task manager and exit there.
as for the in Morrowind folder launch, I have attempted to use the launcher there which brings up the menu to launch the game however, I have to double click it several times to get it to come up at all. this may or may not be related to the crash on exit. I do not think having to click it numerous times is very healthy for windows in the long run, but I am not an expert on how this affects windows but it maybe trying to run a previous instance of Morrowind when I attempt to doubleclick it again. this may relate to the "stopped running" error. but Oblivion does this even after I reboot and try running it. The Morrowind behavior is probably related to how windows handles Vampire Masquerade rather than Oblivion's exit problem, and the start up issue is the same. As with Morrowind, I must, again, double click it at least twice to get Vampire to run it runs after about two attempts but while does not crash on exit it may or may not be affecting WIndows behavior in handling other games ability to run as that was the first time this behavior occurred I don't know why it happened at all.
I used Revo uninstaller to uninstall Vampire and Steam's Oblivion, and Morrowind it did not find any trace registry entries with Morrowind. I found nothing using Glary utilities either. however have used Glary to clean my registry before and think maybe that is the cause of this behavior, don't know.