Do I have to claim and re-check in my baggage in international transit when traveling with the same airline?
I'll be traveling to Memphis, Tennessee form New Delhi, India via Qatar Airways.
The itinerary would be: New Delhi <--> Doha || Doha <--> Chicago || Chicago <--> Memphis
Would I be required to claim and re-check my baggage at Chicago even though all the flights were booked with Qatar Airways or can I check it in all the way from New Delhi to Memphis without re-checking in the baggage again?
Qatar do not fly from Chicago to Memphis. You may be booked on a flight with a QR flight number, but the flight will be operated by another airline, most likely American Airlines (or more likely, American Eagle, which is AA's regional arm). In this case, this does not change the answers already given below.
Yes my flights in US of A will all be operated by AA even though they have a QR flight number.
When you check in at DEL, you should ask that the bag be checked all the way through to MEM, even though yes, you will still need to claim it and re-check it at ORD.
Passengers on all international flights to the U.S. are processed upon arrival at their first airport in the U.S. (including transit passengers; the U.S. does not have sterile transit). In your case, after you land at Chicago, you will disembark the aircraft and proceed directly to immigration and customs screening. You will claim your bag before going through the latter.
If the bag is tagged for MEM, you fortunately do not need to go all the way out to the front counter to check them in again. There will be a station for bag re-check past the customs screening area; you would simply have it scanned there.
Just to add to this, the staff at the airport where you have to claim and re-check in your luggage isn't always receptive to your travel circumstances. I once missed a connecting flight in Charlotte, NC, because it took over 2 hours to get my suitcase and check it in again.
@choster Why does the immigration screening have anything to do with the onward flight? Indeed he will have immigration at O'Hare though.