Kathamo , welcome to FlyerTalk!
I do not believe that Premium Economy is offered by any carrier flying between LAX and JFK/EWR. Each carrier does offer an extra-leg-room section which can be secured at time of booking with cash or miles. Passengers with status can get these seats for free, either at time of booking, or at check-in, depending on status level.
You will not get lounge access with status in U.S. frequent-flyer programs when flying domestically; if you were to obtain mid-tier or higher status with a foreign,
oneworld FFP, you would get domestic lounge access when flying AA or AS, but it does not sound like you would be flying nearly enough to earn such status. AA, DL, and UA all offer high-end credit cards (through Citi, Amex, and Chase, respectively) which include lounge membership. Whether the high annual fee associated with those cards makes them worthwhile for you is something only you can decide.
I'm not sure why you are not considering AA, which flies LAX-JFK and LAX-IAH. (AA's oneworld partner, AS, flies LAX-EWR.) Status in AA's AAdvantage frequent-flyer program can be earned through a combination of flights, credit-card spend, hotel stays, car rentals, AAdvantage Dining, purchases made through the AA e-shopping mall, etc.
Without status, a free checked bag is often a benefit of holding the airline's co- branded credit card. But with AA, that benefit extends only to domestic flights.