Re Applying for H1 when H4 COS is underway(H1 -> H4 -> H1)

My husband is in 3rd year of H1B. He lost his job in August 18 2017 ending his H1B on October 18 2017 (including the 60 days grace period you get on H1B after losing a job). However we applied for his H4 (based on my H1B) on October 3rd 2017 and we got the receipt. Now he got another job and the employer is willing to do H1B. My questions are:

  1. Can the new employer file for H1B without having to wait for the H4 Approval?

  2. Can my husband start working as soon as H1B receipt is received or should he wait until the new H1 is approved?