Congrats. Yes they can send it to you. Keep in mind that lengthy stays outside the US can impact your ability to retain the LPR status.

Please check the requirements of your J1 to see if you are allowed the various job opportunities you are pursuing.

I would talk to your DSO. CPT has strict guidelines.

Medi-Cal Immigrants

You should contact Medi Cal to verify your understanding.

Once the N400 petition has been filed, do we have flexibility with appointments ?

If they have the correct address then I am not sure how one would miss the mail? Also, unless really necessary I would keep the appoints and travel...

Adopting child for a green card holder

You need to talk to an experienced immigration attorney who handles adoption and adjustment cases.

Is child of US citizen (born abroad) in India a US citizen ?

Yes your child may acquire citizenship based on citizenship status. You would need to get consular report of birth abroad from the embassy and then...

Immigration physical exam?

This is not going to be a grounds of inadmission. The DR can of course ask and likely will.

Bringing children to USA

You can petition for your mother. Once she has the LPR status she can petition for her children. The will not be citizens immediately.

I-131 Advance Parole

The cousins wedding is fine for a short trip. Keep in mind that it may take up to 3 months plus to get the AP and while the I485 is pending you...