If "American Airlines" is the name of a company, "go" is correct.
I can't apologise enough, razer, I meant to write "goes".Thanks Bhaisahab and Mohammadhelmi for helping me. In another English forum one person, who has been recognized as ''proficient English speaker'' from that forum, told me this "it's a single company. I'd say "goes" 100% of the time." You agree with his/her logic? It is confusing.
Link to that forum Urgent: go or goes?
I am not sure what the Americans would do with #3. I imagine they'd go for the singular, as 'American Airlines' is the (singular) name of the company.