My flight is from Reagan National Airport (DCA) to John F. Kennedy International Airport (JFK, operated by American Eagle) then to Los Angeles International Airport (LAX, operated by AA) and then to Shanghai Pudong International Airport (PVG, operated by AA). My luggage will be checked through or I need to pick it up and re-check it at one of these airports?
Your luggage will be checked through to China as American Eagle and American Airlines are code share flights assuming you booked it on the same itinerary. If not, you can request them to check it through to the final destination.
Since all the flights except the last one are local flights, you will not need to collect and recheck in your baggage after customs, and it will suffice to do so once you land in China.
Normally yes, you should be fine if you booked the flights as one trip.
Specially if you also check in and get all the boarding passes. If there is a longer stop somewhere and you would want to get your luggage out, you can say so during check-in.
The chances where you cannot check your luggage through are only when you are entering a country and continue on a domestic flight so they can force you to go through customs & immigration and other such occasion.