H4EAD to H1B 3rd lottery to H1B Transfer

HI AniI,
I am looking for some clarification on H1B. My H1B was picked on 2021(last year) 3rd lottery through Company A and it is in processing now. (change of status H4 to H1b)
Currently I am working on H4EAD in company B.

Once my H1B got approved,

  1. If I didn’t even work for 1 day on my H1B, am I eligible for H1B cap exempt later?
  2. Is h1b transfer allowed if I didn’t work 1 day for the sponsored company A ?
    My start date will NOT be in Oct 1 because it was selected in 3rd lottery recently. My status will be changed immediately from h4 to h1.
  3. Do I need to take leave or resign in the company B during the H1B transfer (H4EAD to H1B employer A to H1B employer B) ?
    Please suggest.

Thanks