Yes
No, you cannot stay in the US waiting for green card just because I-130 got filed. You need to have other legal status to be in the US. You won't receive GC if you stay in the US out of status.
Doesn't matter. Those things don't give you legal status.
Many years, probably around 10 if you are unmarried. If you are married or will get married while waiting, it will take even longer. Right now, similar cases filed in December 2016 are being worked on.
Yes, you most likely will be asked for it.
You can say your ID expired. Or you can renew it and show to officer. It's your choice. The more IDs you have, the easier life is in general.
Parole in place is only available to certain undocumented people, primarily spouses, children or parents of US military personnel. Are you in this category? I never heard of anybody getting parole in place. Typically it's people who are out of status for decades with somebody in the military as outlined above.
Interesting, good to know!
The other problem may be preclerance airports - Dublin, Toronto Pearson... The CBP there have a lot more discretion and can reject easier than when you're physically on US soil. I'd keep that in mind too.
And I wouldn't say it's a walk in the park with US CBP every time too. Without original you may be put in secondary if you get particular officer who is not satisfied with copy of extension letter. This could be few hours of not so fun time in a small room.
I would never recommend travelling without the letter.
This is unlikely going to happen, unfortunately. You can meet relatives outside of the US if they can travel somewhere. The plan of your wife going to the US during Christmas break this year is not realistic. The chances of this happening are close to 0.
You need to have at least 18 months total days in the US in the last 3 years to be eligible.
Right now you are not eligible.
So if you don't travel internationally for the next 6 months, you'll qualify to apply. But then I would not travel until you become a citizen because you'd be barely meeting requirements and any more or less long trip can make you ineligible.
My extension letter (I-797) was printed on green watermarked paper. I didn't have online copy, but hard copy (I-797C) wasn't the same, it was black & white and not protected.