If I file Form 1040x because of the Mistake on our W2 will that help my I-751? And how about I and my wife owing the IRS because of the previous money we got from them?

You have to fix this in order to prove that you're married. Filing "married filing jointly", you get most tax credits and every benefits possible. Get tax professional if you don't know how to do it right. Most important, don't mess with the IRS.