Traveling while I90 GC correction is in process?

No, you do not need to remain in the US after filing the I-90 and doing your biometrics.

For I-551 for do we need to file an application first or we can take an appointment directly? how long is the process time?

Your facts are a bit confusing. Why do you want the 551?

Sorry, the fact that he entered unlawfully and is over 21 means that he can not get a greencard without leaving the US.

Contact your US Senator's office ... they should be able to help.

Is there any hopes for overstayed C1D crewman visa?

To be honest, it is nearly impossible for a CiD crewman to adjust status in the US. Meet in private with an attorney.

Do I need to done medical now or wait until get interview letter or RFR letter?

I suggest that you wait until specifically instructed to send/bring ion the medical.

Green card lottery and nonimmigrant visa J1 Trainee

You should be OK ... but it would be best to talk to the company's lawyer.

Would late reply to Request for more evidence leads to denial? ?

As my colleague pointed .. you probably have a grace period. Did you talk to the lawyer?

Made mistakes for N400 form?

Yes, this can be fixed at the interview ... don't stress.

Can I apply for green card with my family?

You need to talk to an attorney via Skype. We don't have enough information ... such as your age and what category that your family used to get...