Do short train rides qualify for the 2for1 deals? I'll be staying near Surbiton station and want to go to Hampton Court Palace.
That is fine, as long as you buy paper rail tickets. The usual advice here is to use Oyster or contactless for such journeys inside the London travel zones.
Neither would qualify you for 241.
A paper ticket is £5.90 return. Oyster etc would be £5.20, but the extra 70 pence you far recoup from the savings at the Palace.
Additionally you can use an eticket providing it has the National Rail logo on it.
Using a SWR Touch smartcard (stored ticket or Tap2Go) would also qualify but not worth the time acquiring one for a one-off.