I have a pending I-130 for parents. They just moved to a new house in the same city. Do I need to report address change?

If your parents are outside of the US, the AR-11 would not be used to report a change of address.

Immigration question. I currently have a green card with no conditions on it.

It may very well affect your application. Consult with an immigration attorney.

Does the husband have to sign the birth certificate of the new born baby in an immigration interview

No. Birth certificates are typically prepared at the time of the child's birth.

How long do I have to wait to get a green card if my wife is a permanent resident?

It could take a couple of years easily, depending on your country of birth.

You should speak to your company's immigration attorney. The DOL will look at "actual minimum requirements" and not simply stated requirements.

I lost my naturalization certificate, but am a US passport holder. I am a sponsor for my father to obtain a green card.

Yes, you can use the US passport to prove your US Citizenship.

Is there a way to get my immigrant visa ?

If you are out of status, you would need Section 245(i) eligibility to adjust your status in the US. Alternatively, you would have to apply for an...

Do I have to be registered with the SSS (Selective Service System)?

Only you can register for the Selective Service and you must be at least a green card holder to do so.

Change of status while assylum case is pending.

The asylum clock will not stop solely by virtue of a change in nonimmigrant status.

Can I keep working with Company A (I do have approved H1B from this company) if my H1B petition rejected, filled by company B?

You could remain in valid status with Company A.