Can I travel internationally to home country without passport?

I'm a green card holder living in the United States. My mom is a naturalized US citizen with a passport, I do not have a US passport because I'm not a US citizen. She became a citizen after I turned 18. We want to travel to our home country in Vietnam in September 2018 for one month.

Can I travel with her to Vietnam with just my green card and Vietnam visa? We did it before 5 years ago, but I'm unsure we can do it now. One airline accepted green card and visa only but another airline wanted passport too.

I called Asiana Airlines recently and asked about my situation. The lady said I might be able to get to Vietnam with just green card and vietnam visa but I might have a problem getting back into the states. She told me to go to the US Embassy in Ho Chi Minh city and talk to the consulate and they might be able to help me out.

Will they seriously let my mom go back to the states but not her daughter? Has anyone ever been in my situation before? Does anyone have a serious answer to this problem? Thanks in advance.

I heard that US airport security only checks the plane ticket where your first stop is and not the entire itinerary. So if my itinerary has 2 stops before heading to Vietnam. Like flight from LAX to San Francisco to Vietnam, I would only have to show my driver's license and not even my vietnam visa. Is that true?

Does snus (tobacco) come up on an airport baggage scan?

I'm under 18 and had snus tobacco in my backpack about a month ago. There's a small black stain from the little packet it comes in on the backpack. I've cleaned it, but it only faded slightly. Will this show up as traces of drugs or something on a baggage scan?

Which airline would you fly from US West Coast to India?

Flight options are listed below. The only flight that would not be in Economy is on British Airways. On that flight it would be in Economy Plus "World Traveller Plus." I'm not sure if this upgrade would make the decision easier compared to some of the other airline options.

British Airways World Traveller Plus

Emirates Economy
Ethidad Economy
Qatar Airways Economy

Cathay Pacific Economy
Asiana Economy

Lufthansa Economy
Virgin Atlantic Economy
Air France Economy
United Airlines Economy