From Cap-Exempt H1-B to H4 to H1-B

I am currently working on H1-B cap-exempt visa with my current employer and it is set to expire in August 2018. I have gotten an offer from my spouse’s cap-subject company who will be applying for my H1-B visa on April 1st 2018 which will be filed as premium processing. I wanted to know my available options to avoid any gap between employment. My spouse’s company is willing to apply for my H4 visa so that i can start working for them until my H1B gets approved and i can start working on H1B on Oct 1st 2018.

Q1. Can i work with my current cap-exempt company till the expiration in Aug 2018 if my H4 gets approved?

Q2. Can i transfer from my current H1-B to H4 and start working immediately with my spouse’s company as his dependent?

Q3. How long does it take to transfer from H4 to H1B when my new cap-subject H1-B visa(premium processing) is approved?

Q4. What is the best way to navigate through this situation so that I don’t have to stay home in between my current cap-exempt H1B visa expiration in Aug 2018 and the start of new cap-subject H1B in October 2018?

P.S. Mine and my spouse’s I140 petitions are approved.