1. Not sure about first question. It appears to me that you can file H1 transfer and start the new job right away, but the final approval will need a quota and won't happen soon. Don't quote me on this.
2. Once you use EAD, you lose either H1B or H4 status. You need to leave the coutry to change back to H1.
3. April 1. The start date of H1 needs to be Oct 1.