firefox doesn't display bookmarks in the sidebar, only in the menu bookmark. It doesn't display bookmarks in the "organize" bookmarks. It doesn't save tabs on exit. Problem began with 3.6.6
FF 3.6.6 doesn't display bookmarks in sidebar, but a search in the sidebar shows they are still there. It does display them from the menu bar. It doesn't display bookmarks in the "organize bookmark" display. It doesn't save tabs on exit. Doesn't save browser screen size.
Alla svar (1)
With so many issues with your current profile it would be better to try a new profile. Be careful with what you copy to that new profile and it sounds that at least you have problem with places.sqlite and maybe prefs.js and localstore.rdf as well. First test that new profile without copying files and installing extensions.
Create a new profile as a test to check if your current profile is causing the problems See [[Basic Troubleshooting|#Make_a_new_profile|Basic Troubleshooting:Make a new profile]]
There may be extensions and plugins installed by default in a new profile, so check that in "Tools > Add-ons > Extensions & Plugins"
If that new profile works then you can transfer some files from the old profile to that new profile (be careful not to copy corrupted files) See http://kb.mozillazine.org/Transferring_data_to_a_new_profile_-_Firefox