When you download something from PS+ a timer is attached, doesn't matter if you disconnect from the internet, that timer and license is stored on the PS3. When it expires, it expires.
As to playing PS+ offline, this is fine as long as your subscription has not expired.
I don't know what you mean by this. I have games installed that i got for free with PS Plus that i downloaded 2-3 years ago and they still work. Some even kept working for a while after my subscription expired.
No they didn't. If you add days to a subscription while you have an existing one then games downloaded prior to the added days may say 'expired' when they reach the original date (before added days), but playing them again will make the display change to real days remaining.
If a subscription expires for real then it expires and no games will work.
Dead Nation, Tomb Raider and some others kept working while i wasn't a Plus member anymore. Also as long as i keep them installed everytime i sign up for PS Plus a few months the games work again.