I can tell you that '10 and '13 versions of Office work fine side-by-side, haven't tested '07 and '10 together. The way I did it was:
[ol 1]
[li]Office 2010 was already installed[/li]
[li]I installed Office 2013 but to a different custom folder[/li]
[/ol]
Now if I double-click any office document/file, it opens with 2013 by default. Some things are better in 2013, but I think some at least look better in 2010, but that's beside the point. One other odd thing I have is that now if I open a spreadsheet with 2013, it'll have 2 windows open - 1 for the main application, and 1 for the open document. I seem to remember having this issue when getting upgraded from 2007 to 2010 at work, but I don't immediately remember the fix, nor if the same fix will work for 2013. I think I posted the fix in a thread where I began asking about it, so I may run a search here eventually when I'm at home sometime. I can still open with '10 or '13 at my choosing. I'm trying to get used to just using 2013 there, at home, and will probably eventually remove 2010, and see if I can get rid of the phantom Excel window at that time. I might even run through another whole clean Windows install to do it, we'll see.
Anyway, if you do uninstall 2007, please report back with success or failure.
If you uninstall '07 and things still don't seem to run right, I'd suggest:
[ol 1]
[li]Use CCleaner to do a general system clean-up, and registry cleaning if you trust it enough (I've never had any serious issues - only seen one minor issue on one teenager's laptop some 8 or so years ago, I think).[/li]
[li]Run a repair of Office 2010[/li]
[li]If that doesn't work..[/li]
[li]Uninstall Office 2010[/li]
[li]Run CCleaner[/li]
[li]Reboot[/li]
[li]Install Office 2010 again from scratch.[/li]
[/ol]
"But thanks be to God, which giveth us the victory through our Lord Jesus Christ." 1 Corinthians 15:57