FWIW, simply setting the default application for .indd files doesn't do what you think it might.
InDesign installs registry entries and a “shell extension” that controls both the icon display for InDesign files (the correct icon for the particular InDesign version) and the program to execute.
Setting for InDesign 9.0.2 (CC) on my Windows system in registry as viewed with regedit:
If you have multiple versions of InDesign installed, only the last version's registry entry survives. If you uninstall any one of the multiple versions, the registry entry is lost and you get the situtation which you describe. Thus, if you uninstall a version of InDesign, you really need to reinstall the version that you want to survive in order to make sure the InDesign registry entries are restored and kosher.
Bottom line is that InDesign really isn't fully setup to support multiple versions installed in parallel under Windows with integration with Windows Explorer.