Before I uninstalled it I got error messages saying Quicktime was not up-to-date, which started me down this path.
Okay ... They aren't classical versions of the symptoms (still, those change on a regular basis), but you're getting enough things going on here to make me suspect that some other application (other than QuickTime itself) has stashed old QuickTime componentry down in your system files. (That can cause QuickTime launch and install trouble, and also those "out-of-date" messages when launching iTunes.)
So we'll go looking for the older QuickTime componentry in the most common locations for it to be stashed.
First we'll need to change some view settings.
In your Start menu, open Computer.
In your Organise menu, select Folder Options.
In the View tab, make sure that "Show hidden files and folders" is
selected, and Hide extensions for known file types is
unchecked.
Click OK.
Now in Computer, open your C:\ drive (or whichever drive you have your operating system installed on).
Open the "Windows" folder.
*If you have a 32-bit version of Windows,* open the "System32" folder.
*If you have a 64-bit version of Windows,* open the "SysWOW64" folder.
What files and folders can you see in there with
QuickTime in the title? (In a standard installation of Quicktime you should be seeing precisely two
files ... QuickTime.qts and QuickTimeVR.qtx ... and no QuickTime
folders whatsoever.)