Here's what I think:
#1 It depends on whether you wish to use your H1B visa or not, since if you want to do a visa transfer then your future employer needs to sponsor it. On the other hand, if you want to switch to EAD for your future employment then you won't need sponsorship any longer.
#2 You need to ensure that you are not deviating too much from your original job category, since as per the AC21 portability rule you'll need to prove that your new job is in the "same or similar occupational classification" as the job described in the immigrant petition filed by the prior employer. I'd suggest that you talk to your new hiring manager and discuss this beforehand, and continue to keep this in mind even as you progress in your career.
#3 The 6-month period is counted from your I-485 receipt notice date
#4 If you don't have it already, get a copy of your PERM labor certification and I-140 approval notice