Yes you can pursue and should. It is not the wrong date which really is bad; everyone makes mistakes. What is really the issue is the cover up...

Do I still need to have an interview and medical exam again?

It doesn't mean anything. The online information is not official and often incorrect. Until you receive hard copies in your hand the other...

How can I apply for residency

You need to see a Board Crtified immigration attorney. And yes you can probably get residence.

I am wondering if a low income student can still hire a lawyer for changing a J1 status to O1A or any other available options.

First, the O-1 is a temporary solution since you will still have the two year requirement. Second, having "a publication" is not nearly enough to...

Does a marriage certificate obtained by marrying an American citizen or a permanent resident give you legal status?

A marriage license does not do anything for you except allow you file for certain immigration benefits. But until you file for and receive those...

Assuming the CPT was all full time, CIS may be correct. If any of it was part time they are not correct. This decision should have no impact on...

What is my option after getting divorce and got deportation letter from immigration

You need to consult a Board Certified immigration attorney. I assume you are out of status now so the only option is marriage to another US...

Vawa petitioner

Since it was denied you should at least discuss with another lawyer, preferably one who is Board Certified. You can email the denial first so we...

Applying Green card for spouse

It is really important that you consult with an Immigration attorney immediately. These are very technical issues which cannot be handled by...

Failed to update my address

Changing the address will not be a problem for naturalization.