A corrupt font does seem to be a likely possibility, and reinstalling it is pretty easy from the Office CD.
Times New Roman should be in the /Library/Fonts folder (as well as under the Microsoft Office X application folder). If you also have it in your
/User/username/Library/Fonts folder, that would be a duplicate and I suggest you remove it, remove the Office Font Cache (10) file again, and retry Word.
You say documents using Times New Roman still displays in existing Word documents. I'm curious what the Toolbar Font drop-down menu displays when Times New Roman text is selected (that menu usually displays the name of the current font.)
If you select Customize... from Word's Tools menu, go to the Command tab of the Customize dialog, and select Fonts in the Categories list on the left-hand side, does Times New Roman then appear in the Commands list.
Word generates the Font menu and Font toolbar pull-down menu content automatically from available Fonts, so you can't customize it. If Times New Roman shows up in the Commands list of the Customize dialog, you could drag it to the toolbar as a work-around so you can apply it.
(I made a custom "My Fonts" menu (starting with the New Menu Category under Customize commands) in my toolbar with my top ten fonts, so I didn't have to scroll through all the fonts I rarely use to access my usual choices.)