TextEdit shows dialogs about not having permission to open any files
I ended up replacing TextEdit's code signature with an ad-hoc signature:

sudo codesign -f -s - /Applications/TextEdit.app/


It disables sandboxing, so for example the preference files are in ~/Library/Preferences/ instead of the sandbox container.

Edit: The dialogs returned after I reinstalled OS X, and now I'm getting errors like this for codesign -f -s -:

$ sudo codesign -f -s - /Applications/TextEdit.app/
/Applications/TextEdit.app/: replacing existing signature
/Applications/TextEdit.app/: object file format unrecognized, invalid, or unsuitable


I'm using https://github.com/jjgod/TextEditPlus for now. It's based on a version of TextEdit that came with 10.7, but it works with 10.8.2.