Nope you cannot :-

"My grace period ends on 10/10/2010... Do I have to leave US ?" - what do you mean by grace period.. It does not matter that you have a grace period by the end of you OPT you have to have something in process besides H1b which starts at 10/01 ususally. But after 8/10 you need to either apply for college that starts in Sept or apply at a university with quater system that starts at Nov.

Your best bet is to apply for the college with quater system. you know for sure what will happen to your h1b.

"My H-1b gets approved on 10/20/2010... Can I stay in US because I am in process of application for H1-B?"

Once you get you approval then only you can forego your status besides that you have to maintain it everytime