If you are connecting through to Paris via London
1) No.
2) If it's on one ticket, then your luggage should automatically be transferred to your Paris flight. Make sure the tag the airine puts on in the USA says CDG or ORY depending on which airport you are going to in Paris.
3) Many Americans have the bad habit of calling both immigration and customs.....customs. It's a mistake They are two different things. Customs is for your goods, your luggage. Immigration/Passport Control is where they check your passport. You won't do that in London. That will be done in Paris.
Normally, when you arrive at your destination you get off the plane and go straight to Immigration/Passport Control. After you clear that, you are directed to the baggage claim area. After baggage claim, you go through customs. (Usually just a walk through.) Then you enter the main terminal area and can leave the airport.
4) All transfer passengers go through security again at Heathrow.
Follow the link Joe gave above and it will give you step by step instructions.