I wanted to share my experience purchasing and using a 7 day London PAPER Travelcard, as online information is scarce and somewhat confusing.
The 7 day PAPER Travelcard (NOT the 7 day OYSTER card) can be used for the Days Out Guide 2 for 1 promotions. The paper ticket has the National Rail logo that looks like a chicken foot. You can purchase the 7 Day Paper Travelcard at ANY National Rail station WITHIN London (Paddington, Waterloo, Kings Cross, etc.). You DO need a passport size photograph. When you purchase the Travelcard you will receive a typical tube style paper ticket good for 7 days (good on all buses, tube, DLR, etc. within Zones 1 and 2) and an ID card to carry with you (the ID is to theoretically prevent you from sharing the 7 day ticket and is created on site by the ticket agent using your passport photo. Your photograph is essentially taped to a pre-printed ID card with super adhesive non-removable tape). You will also receive a guide listing all of the Days Out Guide 2 for 1 promotions. This guide also contains the "vouchers" in the back of the guide for you to complete prior to your arrival at the sight you wish to visit. Have a pen handy. You can also go onto the Days Out website and download vouchers prior to arriving in London.
At the Days Out participating sight, present the completed voucher and your Travelcard ID (though I found most ticket agents at the sights barely glanced at the ID) and receive your 2 for 1 tickets.
That's it. The paper ticket works like any other "1 time use" ticket you would purchase prior to entering the tube (so don't lose it as it is non-replaceable) and the Days Out promotion is easy to use and saves a significant amount of money.