Switching from L1b to H1b

USA H1B Visa
1

Hi
I have L1b from Company1 and also had a H1B from Company2(Future employment). I recently Interviewed for a Company3 and got placed and Got my H1B transferred to Company3(i.e,Company2 H1B transfer to Company3).

  1. I Can directly give notice period to my Company1? will it cause any issues with company 1 ?
  2. If I resign from company1 I can directly start working from company3 without any issue ?
2

Yes, no issue.

Yes, no issues.

3

So If the Company1 with L1b will just cancel my L1B Visa that’s all right? “They will not ask me to go to India or anything on those lines right ?”
and
Once I move to Company2 on H1B visa do i need to update to any other agency in US ? or it is taken care by company ?