I believe you should work at least one day

in order to be eligible for h1b portability so you are not subjected to h1b quota.

if you don't care about the h1b transfer fee, you can always try. i wouldn't file h1b transfer in your case if i have't to pay myself

if you work at least one day on or after oct1, i'm not sure if you still need at least one pay stub, maybe not. You can start to work at company B if you are sure cis receives your case (maybe no receipt notice, but a case number from cancelled check)