No
Yes
Definitely yes. Done that.
Ask your lawyer. If you are on h1b and you are changing employer, you can continue current job until the transfer is done. In fact you have to, there is no other option without going out of status or country :)
How long it takes for H1B transfer?
Yes of course
Yes, you can work while your H1B transfer is in process. I am in same situation, it’s always better to resign once you get H1B transfer approved. No Issue, keep working for your current employer. You have valid status with your current employer.
Yes