Marriage based green card - will it trigger fraud or red flag if the US citizen has petitioned multiple times for spouse?

Yes this can definitely raise red flags. I would be sure to gather as much evidence to prove that the marriage is legitimate.

How can I go throught the process of Natrualization properly based on my circumstances after being denied previous application?

There are not enough details on your case. As a general rule, you need five years of good moral character to be eligible for naturalization. If...

I move to another place, after submitting the papers

You can go to uscis.gov and change your address online.

Is coming to the USA under different name considered Entry with Inspection or Entry without Inspection?

Any time you use fraud to enter the United States it is considered an entry without inspection.

Is it possible to get an ice detainer lifted on a canadian citizen being held for illegal reentry.

As a general rule, there is nothing you can do to lift the ICE hold. They will probably just reinstate his prior deportation once he is back in...

I don't have all the facts of your case but since you are a citizen and she was at one point admitted as a lawful permanent resident, you should...

Does having a baby help me in allowing me to stay in the u.s. while filing for permanent residence?

No that will not help you.

Removal Proceedings/marriage

Definitely apply before. That way if it is approved before May you can ask that the Judge terminate the case so you can apply with CIS instead of...

Legaly staying in U.S

Yes. You can change your status to a F1 student. Go to uscis.gov and look up "Change My Nonimmigrant Status." Good luck.

Naturalazation interview

Well you should definitely send in what she requested because if you don't the application will definitely be denied. Also make sure that you get...