Was that a clean install of Mavericks or an in place upgrade?
It was an in place upgrade.
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In place operating system upgrades are nothing short of a crapshoot. My recommendation is to back up your data and do a clean install.
OK, while I have your attention. I successfully installed InDesign again.
The document I am working on contains an illustration supplied by the client
and when I try to select or delete the illustration, the program crashes.
Try exporting the document to IDML and opening that. That usually clears out any corruption.