Yes you should be able to move and carry proof of filing and PFD.

F1 visa?

When will you become a USC? It is better to file for her then and adjust status rather than as a LPR where you will need to wait under the F2A visa...

What is the affect of divorce on my Temporary Green Card Status?

You can file for a waiver of the joint filing requirement either through divorce and a showing of your good faith marriage, through abuse that...

My I-130 is pending for my wife but her I-94 will expire soon. Do I have to submit I-539 ?

Filing an I539 at this time with a pending immigrant petition would be meritless given the contradictory intent you have established.

Wife pregnant from her ex boyfriend, can I still get the green card?

With these facts, an approval will be difficult and it is best to prepare for an interview with an attorney, after full review of your case.

Likely not but you should review your lease to confirm this.

My wife needs to sponsor me but she is unemployed?

You may use a co-signer for this requirement who is a USC or LPR and meets all eligibility requirements.

My wife is unemployed and has to sponsor me for i-130.

The I130 does not require sponsorship, rather if you are applying through consular processing, at the NVC phase you will submit the I864. You may...

It is possible as it is a question that you answered on the N400 but should not impact your case.

How long can you stay out of US if you are permanent resident?

Remaining outside of the country for more than 6 months may present problems for reentry or continuous residence for naturalization purposes. It is...