The answer is "it depends". You get free access to the Admirals Club if you're flying in First or Business class on a "qualifying" flight. And the flights that qualify are international flights (but not Canada or the Caribbean) and three-class non-stop transcontinental flights in the USA. So not most flights in the USA. For details, see https://www.aa.com/i18n/travel-info/clubs/admirals-club-access.jsp The upgraded lounge that Roberto mentioned is called a Flagship Lounge. Those have the same access rules that I listed above but they are only available in ORD, DFW, LAX, MIA, and JFK. Coming soon to PHL and LHR.
I'm pretty sure you can, in fact for first class tickets there is an upgraded Lounge for first class tickey holders. Ask when you get to the entrance of Admirals club. That is a separate Lounge, inside the Admirals club, but only in certain big airports, for example Miami Airport. Always ask!
I think in some u can. I am not sure
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