Teemo

I don't agree with this as a definitive statement, since it was approved already as opposed to people who leave the US before it is approved and abandon their application, but yes in this case it carries some risk and I wouldn't do it. What if the USCIS messed up and hasn't actually approved it yet? What if it gets lost in transit to your wife or to you in London? Risk far outweighs reward here. Reschedule your trip. 

You absolutely should not do this cruise without advance parole. It is definitely considered leaving the country and you will be deemed having abandoned the AoS application and likely won't get back in. Luckily it is still more than 2 months away and you applied 2 months ago so you might get it. Took me 120 days to get my AP

I don't agree with this as a definitive statement, since it was approved already as opposed to people who leave the US before it is approved and abandon their application, but yes in this case it carries some risk and I wouldn't do it. What if the USCIS messed up and hasn't actually approved it yet? What if it gets lost in transit to your wife or to you in London? Risk far outweighs reward here. Reschedule your trip. 

Never make plans you aren't willing to change or cancel until you have the document in hand.
 
Personally, I'd wait for it to arrive to be safe.
Technically, under the letter of the INA, one must only be approved for AP and use AP to re-enter the US as a parolee. They do not need to be in physical possession of any document upon exit (just have it approved).
But again, it's not something I would prefer to deal with after coming so far along in the process.

Yes, you are going to go to Canada, of course you need a passport. And you need AP. 
 
Even if you don't get down of the ship, you will be in Canadian waters and international waters. 

You absolutely should not do this cruise without advance parole. It is definitely considered leaving the country and you will be deemed having abandoned the AoS application and likely won't get back in. Luckily it is still more than 2 months away and you applied 2 months ago so you might get it. Took me 120 days to get my AP

With AP, an overstay is irrelevant.  Plenty of people with overstays have successfully used AP to visit and return to the US to continue their AOS.
 
Marriage to a USC does not prevent someone from getting a ban.  Plenty of spouses of USC have bans.

The only reason you could end up be refused entering the US with AP is 1) your AOS is denied while you are out of the country, or 2) you do something that makes you ineligible to use AP - i.e., commit a crime while outside the US with AP.

Only USCs are guaranteed entry.
Being married to a USC does not affect whether you get a bar upon exit or not.
Exit with a valid AP will not trigger a bar due to unlawful presence.
Bars start at 180 days of unlawful presence.
usually F-1 applicants are admitted as D/S, so you would not start accruing unlawful presence until they have determined that you are in the US unlawfully.
A pending AOS applications tolls the unlawful presence accrued (so you don’t get more while it’s pending).

The 3 to 10 year bars only trigger is you overstayed by more than 6 months or 1 year respectively. Now with that being said even if you did overstay by 20 years it doesn’t matter anymore. The Bureau of Immigration Appeals finally ruled in 2012 (Matter of Arrabally and Yerrabelly) putting an end to this. As of that date travel with advance parole is no longer considered a “departure” from the U.S. and in essence can no longer trigger a bar after leaving the U.S. briefly. Before then leaving even on AP was considered a departure therefore triggering any unlawful presence the alien accumulated. With that being said parole is not admission and is at the sole discretion of a CBP officer so it carries a small risk, however; provided you don’t have any other inadmissibilities such as criminal ones you should be fine. Many on this forum have used AP in similar situations since then and returned just fine after a few questions. If you really have to travel go ahead you won’t be the first. By the way 1 girl here travelled after overstaying for 7 years on a tourist visa and returned. All she was asked were a few questions and paroled in. 

You are spreading so much disinformation!!!! A pending I-485 puts you into stay authorized by the Attorney General. He's kind of a big deal when it comes to immigration enforcement. So, no, you are not subject to deportation unless something changes like you commit a  crime.

No to social security card you have one already. That designation is not removed until you are a US citizen.
 
F1 student.
 
I hope you realize you cannot leave and re-enter the country until you have received your advance parole? If you do you will be abandoning your application, and will have demonstrated immigrant intent with a non-immigrant visa (F1) and therefore will have a tough time getting back in. 

Hmm. I haven't really heard of people being denied AoS AFTER they were granted EAD/AP but BEFORE the interview. There is no extra information on which to be denied on. 

This is not correct. You are not in lawful status once the I-485 is received, unless you are in lawful status from your original I-94. (for instance your H-1B is still valid). The I-485 places you in authorized stay from the Attorney General, which is not the same as lawful status. If you are a visitor overstay, you do not have lawful status now, you are just in authorized stay, which is a different term. 
 
When you receive your EAD, you are still in authorized stay, and it is lawful for you to work, and with AP, you are allowed to exit the country and not abandon your application. But you are not in lawful status as per the definition of that term. 
 
Your work authorization document, while not giving you "lawful status," is proof of your authorized stay and ability to work. This therefore entitles you to a state driver's license. Just take your EAD and social security card and whatever else they require and you can get a driver's license. 

How so? How else would you expect them to use a background check? 

I don't think you understand reality.
 
By federal law, all employers are supposed to ask for work authorization to hire you. And they are supposed to have you fill out an I-9 to prove so. Many of them do not, especially the smaller ones. How else do you think 11 million undocumented immigrants are working without authorization? Some falsify documents, but many more are not even asked to prove they can work legally. The local deli on the corner or the landscaping company with 3 contractors rarely ask for work authorization, so they are hiring people who should not be working.
 
Now, if you work without authorization because the company has not asked you to provide documents, you have unauthorized employment. This is a serious issue - EXCEPT for people who then apply for adjustment of status through immediate relative that is a US Citizen, or have already applied for it, because that is forgiven. So he is in the midst of adjusting status and assuming he get it (this won't be a bar to it), it will be like it never happened. So what possible consequence are you envisioning, that millions of others have not received? Are you conflating the consequences for misrepresenting oneself as a US Citizen or presenting false documents? Are you conflating the serious consequences for unauthorized employment that those applying for EMPLOYMENT-BASED not FAMILY-BASED visas will face?
 
Nothing here is meant to condone or encourage unauthorized employment. Simple fact: millions of people work illegally and most of them can do so because they are not asked to prove they can work here. Many of those millions are overstays for 5 or more years and this how they work. Simply because they are not asked to prove they can work here. The second they are granted  adjustment of status, unauthorized employment (NOT misrepresenting oneself as a US citizen or presenting false docs) is forgiven which by definition means all potential consequences are removed. And unauthorized employment is not a bar to being granted AoS. 
 
Employers face the potential civil consequences, but in reality the government does not have the resources to go after small employers, so they audit the bigger ones (Wal-mart, 7-11, Bank of America etc). That is why smaller employers don't feel forced to use I-9s, even though by law they should. 
 

Correct. But all unauthorized work is forgiven for immediate relative of USC. 
 
Correct. The employer takes on the potential consequences, which range from $1,000-$2,000 per instance for a first occurrence. The employee does not take on this penalty.
 
Correct. No penalty. No impact on adjustment of status. No legal issues. No liability. Therefore, no consequence. 
 
Correct, but no one said anything about a false claim to US Citizenship. that is not in the discussion here. It is simply working without authorization, without misrepresenting oneself or presenting false documents. In OP's case he was not asked (yet) to prove his work authorization. This happens all the time, especially with smaller employers. 
 
Correct. Nothing I have said here condones or encourages unauthorized employment. But it is a FACT that it is forgiven and it won't impact the AoS, as long as there is no misrepresentation, which OP never said he was doing. All he said he was doing was producing his social security card when asked to do so. 

In trouble for not working? He is not misrepresenting himself, he is not presenting false documents, he is not lying. If they don't hire him because he has no work authorization, that's it, there's nothing to get in trouble for because nothing has actually happened.