Hello Mohit,
1) OS X Lion 10.7.5, for the crash log where would I find it? Or are you asking what the pop up screen says when AI closes, if you are it says "Illustrator unexpectedly quit while trying to restore it's windows"
2) The AI screen comes up once and then it crashes quickly after
3) I have not tried launching AI after disconnecting the internet but I will try that right now and get back to you.
Thanks!