What should I do for 1 month gap of Sept 2021 as my L1B expires on 31 Aug 2021 and if I get H1B approved?

This precisely why you have your own immigration attorney to advise you. That attorney is being paid specifically to advise you on these issues....

How long may I stay without an EAD card or receipt#? Are there any ways to expedite this case?Can I apply for STEM extension?

This what happens when students wait too long to apply for OPT. You will simply Jae to wait until CIS catches up.

Can you register a trademark on an L1 VISA for use in the future, or is that considered work?

Registering a trademark is not considered work or employment so it has no impact on your immigration status.

These are good questions for you to discuss with the immigration attorney who is representing you, and being paid to do so. Good luck.

What are the steps to move my husband case from the dominican Republic to US?

Unfortunately, we need much more information such as what type of visa, his immigration history in the US, the details of the petition, etc.

Can we reapply for a visa?

An appeal will not do any good. You need to refile and also prove-up the validity of the other marriage. Finding the ex-wife may or may not help....

Of course you need to tell your attorney. You have a 60 day grace period so you were not out of status.

I am us citizen need visa to enter in india because my wife is in india witing on us visa ?

That is a question for the Indian Embassy in Washington D.C. Good luck.

Trying to get an F-1 visa in your situation is a perfect way to not be allowed in the US. You are getting very, very bad advice. Do NOT do this.

It is actually a change of status which your employer (for the E-3) will file for you. It is assumed you are not subject to the J-1 two year...