v10.6.8 is the latest OS X version available for Snow Leopard.
As far as not being able to sign into the App Store ...
- Apps such as Little Snitch, Growl, and MacKeeper can prevent apps from downloading
- Disable anti virus software if installed
- Make sure the firewall is turned off in System Preferences > Security & Privacy > Firewall
or System Preferences > Sharing > Firewall