H1b transfer from cap-exempt to cap-subject

I work for company A (non-profit institution) on H1b visa (CAP-EXEMPT). I got an offer from company B and they have filed my CAP-subject H1b this year 2022 and it got picked in the lottery and got approved as well. The start date on I797 (Company B) is October 1st.

Q1: My cap-subject H1b was filed under H1b Change of status, so my status will change automatically from October 1st, in this case do I need to join the company B from October 1st, 2022 to maintain my status or is it possible to start working in Jan 2023, as I’m planning a trip to India in Dec 2022 and till then I was planning to work for Company A (CAP-Exempt) employer.
Does my CAP-exempt H1b status at company A would become invalid automatically on October 1st and I won’t be able to work for company A after October 1st?

Q2: If I have to join Company B in October 2022, can I use my old H1b stamp (CAP-exempt) for returning to India in Jan 2023 or I would need a new H1B stamp (CAP-subject).

Q3: If I can join the company B after I return from India: In this case I’m still employed by company A, so while coming back from India at the port of entry, can I show them company A’s I797 approval notice and it’s H1b visa stamp (CAP-exempt)?

Please let me know if any of the above questions is not clear. Thanks for all the help.

This is fine. Even if the cap-subject H1B is approved with I-94, you can differ to join the cap-subject employer at a later date as far as your cap-exempt H1B I-94 stays valid to keep working at your cap-exempt employer.

No.

Visa is just a travel document and as far as I know, H1B visa class is same for cap-subject vs cap-exempt, so yes you can use your existing H1B visa to travel back even after switching employer provided the visa remains valid as of the date on entry to the US.

Yes, you need to use the I-797 for the employer you wish to work for immediately after entering the US.

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Thanks for answering my questions Kalpesh.

For Q1: I was reading a blog by an attorney where she mentions about a case in which an employee is on L1 and then his H1b got approved with change of status (COS) with October 1st. She is mentioning that when H1b is filed as COS, then the status of employee gets automatically changed from October 1st and the L1 status will be invalid from October 1st. If H1b was filed as consular processing then automatic change of status is not applied and the employee can continue working on L1 until he gets stamping or apply for COS.
This case seems similar to H1b CAP-Exempt to H1b CAP-subject (COS).
Here’s the link for that blog: Change of Status L1 to H1B Grace Period – Rules, Time after Oct 1st? DOL or USCIS ? - RedBus2US

Do you think it’s correct information?

L1 and H1 are totally different class, so I dont think this applies to your cases. Please consult your immigration lawyer and ask them.

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Thanks Kalpesh!
I spoke to couple of immigration lawyers and they have mentioned the same as you said.