Can he travel and what to do now?

You are correct your husband will not be able to come back inside the US with the EAD/AP card. Please consult with an immigration attorney on this...

If you have applied for and received your EAD card then yes you can legally work for anyone you like but if you still did not receive your EAD card...

Birth certificate

It is your decision to make since this is a marriage based green card application, if you are filling out the application with your adoptive...

H1B to L1A conversion while staying in US

The firm's immigration attorneys are in the best position to answer this question.

What address should my new petition be sent to if I am currently living in Hawaii and my petitioner is in CA?

If you are going to be living in hawaii for more than two months then you should update your current address. All the best.

2.Can i change my status back to h1 when i get a new project when his h1 is still in process?

1. Yes you can do the COS 2. Yes All the best.

Can I do a non-study related part-time job when on OPT?

You can work as many hours per week as you wish and it doesn't necessarily have to be related to your field of study. All the best.

Can i file a h1b transfer to a new employer before april 23rd and can i stay in USA ?

You should initiate the process as soon as possible at the latest before April 13th. All the best.

Yes you can apply for the COS in premium processing. All the best.

Can I get h4 ead directly from OPT?

Any process you do the EAD card has a processing time of 90 days. All the best.