Uninstalling/reinstalling dont help much if you use same profile. Make a new one to start over.
You use cookie blocking, like from Spybot (immunizing)? Sometimes I hang out in Firefox irc and quite often people complain about Firefox not remembering cookies, logins. 1. There was a bug about that, fixed in 3.07 2. Many times problem is corrupted sqlite files, like the one holding cookies. Using SQlite manager, an extension you can chech integrity of sqlite files. If corrupted you have answer. Some say sqlite cant become corrupted unless touched by 3rd party programs, like Spybot. There are debates about that
Have no idea but a damn problem because hard to fix unless you know this.
Under privacy both cookie boxes are ticked? Kept until expire? and nothing in exception list? If yes, yes and no I think you could have corrupted file. Sqlite manager can tell in under 1 minute. No extension or userscript messes with Google I assume.
Sqlite manager
https://addons.mozilla.org/en-US/firefox/addon/5817 at right side select cookie.sqlite, press go. In database menu chose Check integrity. If result not "ok" there is problem.
Can a cookie become corrupted? Dont know but if sqlite dont find errors may be try delete all Google cookies, click "Show cookies".