My situation is also same. My OPT expires on may 31st. But i am going to send new i-20 (starting from june 1st) within this week to my lawyer so that he can send this along with other documents.

So my questions are:

1. If my H1 is approved will they send new I-94 along with that or do i have to apply for that too?

2. I am applying for master's quota. Any of you guys know what is the minimum salary requirement to get     
    approval for h1-b visa?

3. And also is it fine to send new I-20 while we apply for h1-b visa to bridge the gap between june and oct?

Thanks