You can start working once the USCIS has received the H-1B petition. It usually takes about a week for the receipt notice.
Yes, that is possible.
There are certain circumstances in which you can change employers. It depends on whether your I-485 application has been filed and is still pending.
This does not happen automatically, as the USCIS does not cancel your L-1. Your employer would have to withdraw the L-1 once you leave.
You get a 60-day grace period upon expiration of your F1-OPT to stay in the country. However, you cannot work during your grace period.