In my reading I found the 2 for 1 offers for London attractions. Can you use these offers even if staying in London if you each possess a train ticket? If I purchase a travel card in advance does this cover the 2 for 1 offer? I see it's possible to purchase this prior to arrival and it includes tube bus and train? Lastly, which Zones should I purchase if only staying in London (not to Heathrow)?
If you are just staying in central London, you only need zones 1 and 2. Whether you choose a Travel Card or Oyster depends on a couple of things such as length of stay and how frequently (times per day? days per week?) you'll use public transport. This site has comprehensive info on public transport and detailed information on Travel Card vs Oyster: https://www.londontoolkit.com/briefing/travelcard_oyster.htm
To take advantage of the 2-4-1 offers you need a PAPER travelcard (not an Oystercard) purchased from a Rail station (not a tube station)...you will need a passport-sized photo when purchasing one.
Note that while the 2 for 1 offers are a great deal, they only work for an even number of travelers. They are not "half off," but two for one. Since I was going alone, I couldn't take advantage of them. If you do have an even number, they're probably worth the slight extra hassle of getting a paper Travelcard from a train station (not a tube station). Otherwise, just get an Oyster Card, and load it with either Pay As You Go (PAYG) money, or if you're in London for 6 or 7 days, you can get a 7 day, two zone Travelcard loaded on to your Oyster Card (that's what I did).
It also depends on your itinerary. If you have a National Rail ticket to London, and one leaving, you can use this pair to qualify. Heathrow Express and Eurostar tickets don't count, but ones from Gatwick would.
More info, also covers "decoy" tickets:
https://www.tripadvisor.co.uk/Travel-g186338-c133479/London:United-Kingdom:Paper.And.Oyster.Travelcards.2.4.1.Offers.html
A key (to me) point: To take advantage of a 2-4-1 offer, you must go through the regular ticket line at the tourist sight. For highly popular sights (including the Churchill War Rooms) the ticket line is often extremely long. You might reasonably decide, as I have, that standing in line for over an hour to avoid paying one entry fee is not a good use of vacation time. There may be other 2-4-1 attractions where you could run into a long wait to make use of the 2-4-1 deal.
Also note that some sites (including the Churchill War Rooms) sell online tickets at a slight discount. That somewhat reduces the value of the 2-4-1 offer (since you must pay full price for the one ticket you buy), and makes me even less likely to want to stand in a long line.
To be clear, Oyster is an electronic wallet rather than a card. It holds money to pay for one-way fares, automatically calculated and deducted. It has a daily limit and after that is spent the subsequent rides are free (for the remainder of the day). Some sorts of travel deals may be compatible. It's the speediest way to enter and exit busy Tube stations. Transport for London explains it all: https://tfl.gov.uk/
More info, also covers "decoy" tickets:
As the article itself says this is unmaintained old information and the 'decoy tickets' of the kind mentioned in it are no longer acceptable for the offer. For Central London you would need a return ticket to London Terminals, or some specific nearby station.