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.