H1 transfer

My husband is working for a company A for the past 1 and 1/2 years. The company has sponsored H1 and H4 for us but we haven't got it stamped in our passports. Now my husband got a job with company B and they are ready to do H1 visa transfer. My query is:

1) How much time it needs to get H1 transfer and till then should he continue working at company A?
2) Is there any reason why the transfer won't take place and he will lose both jobs?
3) Is not having H1 stamp on passport cause any problems for H1 visa transfer?

Please help fast. Thanks a ton.

Yes, the other company can apply for h1 transfer.
The time taken to receive a approval doc is now a days vary from 6 months to 10 months. IF apply via premium procesing with adtional 1000$ fee it can be recived in 15 working days.
One can start working for other company on the baisis on receipt of the H1 transfer filing.
There should be no problem of working without H1 stamp.
Visa stamp is only required if you go out of contry and want to enter back.

Thanks for prompt reply. Please answer my second question too which is

Is there any possibility why the transfer won't take place and he will lose both jobs? To be more clear will he be able to work for company A till he gets visa transfer from company B?

He should submit his resignation after he gets his receipt number from USCIS. As long as company A doesn't cancel his H1b, he is eligible to work for company A. I think they usually do when one gives resignation. Hope this helps.