Hi CM
I used the 2 for 1 the last time I was in London and it's not difficult at all. There are just a few things to keep in mind...
First off, the Travelcard you need is a paper card issued by National Rail, NOT the Travelcard you can add to your plastic Oyster card (although both cost the exact same and allow you to use the exact same transit options). The only way you can get the 2 for 1 discounts is by having the National Rail card, so yes, you do need the NR Travelcard if you want to take advantage of the offers.
Second, I never noticed specific times on anything we did, but the Days Out site will have detailed information about each individual offer and any restrictions. Do be aware though that many offers are only available at certain times of the year, so be sure to check the offers you want to use are still going to be valid during the time you'll be there (we JUST missed out on a 2 for 1 for the London Eye).
Third, you will need to print out the offer and present it when you show up to the site (Tower of London, Hampton Court, etc.). There may be a smart phone option, but I'm not certain about this - best to print it out to be safe.
As for negatives, if you don't mind a trip to a National Rail ticket office (which are easy to find throughout the main tourist area of London) and you don't mind carrying a little extra paperwork, I can't think of any negative to saving a fair chunk of change on some of the more expensive sites in London.
