[eluser]TheFuzzy0ne[/eluser]
Welcome to the CodeIgniter forums!
Interesting. I doubt it is anything to do with the framework to be honest. Users being logged out randomly could have something to do with the session settings (although I doubt it), or that he hasn't encrypted the cookie, and users are able to exploit this, but as for user's being able to gain access to other users' accounts, I'm pretty certain that has nothing to do with the Framework at all. CodeIgniter doesn't handle authentication (because it has no authentication library), so it's up to the developer to write a library or to use an existing library. If CodeIgniter was so terrible, why was your developer using it?
Granted, CodeIgniter does have a few bugs, but certainly nothing that would cause the problems that you're describing. I think that your developer needs to man up, admit that he's a complete idiot, and do something about it. Any developer worth his salt wouldn't knowingly write an application based on a framework that "renowned for having many problems". That would be like building a block of flats on swampland. It's just absolute idiocy and dickheadishness.
Please feel free to PM me. I'd be happy to discuss this further with you, and to take a look to see if there's anything I can do to fix it (provided it's something simple and doesn't take a complete rewrite).