As noted, the episode 'missing' from the third series isn't totally missing, it's merely missing from its original format, unlike the rest of the series. There was also a radio version of 'Dad's Army' and the episodes missing from the television version were performed in the radio version and still exist in that format. It is possible that the five episodes missing from Series Two were never telerecorded as, at the time, Series One had not sold well and it was felt Series Two wouldn't sell either. The episode that does exist from Series Two ('Sgt. Wilson's Little Secret') was recorded directly onto 35mm and still exists in that format. It is unknown (by me at least) if this is the case or not. If it is true, it would make recovery of these episodes very difficult indeed.
The first two episodes of the second series of 'Dad's Army' were officially returned to the BBC on June 1, 2001. They had been sitting in the EMI studio at Elstree along with a number of other film cans for years and were eventually thrown out. An employee retrieved them from the trash and ended up passing them along to a friend. The friend saw the recent BBC appeal for the return of missing material and offered his items. The episodes were at the EMI studio due to their being given to Columbia Studios in 1969 in an effort to interest them into making a 'Dad's Army' movie. One of the film cans even contained a note to return these to David Croft, one of the creators of the show.
The theory that the second series could be difficult to locate due to a lack of popularity has been disputed. Sales of the series were not particularly lower than any of the other existing series.