H1B Transfer while on the 60 days grace priod

I’m currently working with company A and my last working day is Jun 2nd. In the mean while I have got another employer who will be filing my h1b transfer by next week. I hope that I will get an receipt notice by 2nd week of June(say June 15).

  1. Can I legally stay in US from June 2nd till I get my new petition receipt ?

  2. Can I start working for the new employer once I get my petition receipt notice or should I need to wait for the approval?