help about changing job from nonprofit to nonprofit

I am now working in a university under H1b. If I got a new job offer from another university starting from next August and my boss agrees that I leave on June.

1. When should I start to apply for a new H1b or transfer my current H1b? In my situation, which one is better, new h1b or transferring old one?

2. During the gap between June and August, can I stay in the US? Is their any grace period after leaving my current job?

3. What will happen if my boss layoff me before I got new h1b (after I got a new job offer)? What should I do to keep my status?

Thanks in advance!