Can I continue working with my current employer after H1B tranfer ?

Company K has sponsored my H1B visa and I am switching over to Company Y which is sponsering my H1B visa. I have folowing questions :

  1. When I need to resign with company K ? After the H1B transfer to company Y or when the transfer process initiates ?

  2. Can I continue to work with both company K and Y after my transfer as I would be having 2 petitions approved ? Can it be say part time with company K and full time with company Y ?

  3. Can I switch back to company K again in future after working for compnay Y for few years ?

  4. Can company K sponsor my Green Card even if I am not working for them when GC is applied ?

Responses to these questions is highly appreciated !

Thanks !