The curved-t Topps logo debuted in 1966 but I don't believe the little registered trademark symbol was added until 1967 nor did these style packs debut until then. Given that and the slightly psychedelic lettering, I would place the date at 1968-69.
There are 24 known subjects, skip numbered to 36,which look like this in their folded state, as per a Legendary auction a while back:
You would unfold the "card", which was made of paper, to reveal a funny response. For instance, the "I got up to 100 MPH in my sports car" comment is followed by "...the day I saw a fire breathing dragon, as this illustration from Chris Benjamin's The Sport Americana Guide to the Non Sports Cards shows:
I've resorted to a b&w scan as these are scarce and I don't own one. I had to clip a reverse extract from Chris Watson's Non-Sports Bible:
Yes, they are numbered and examples are known up to #36, which appears to be the end number based upon the above scan mentioning "30 of 36". Bob Marks wrote a great article in The Wrapper #68 about the set and noted original artwork was known for both issued cards and unissued ones. The amount of unissued artwork led him to believe there were more than 12 additional subjects being planned but if the first series was to consist of 36 only, that means a second series must have been contemplated. Marks also noted the "answer" flap was part of the reverse as issued, lightly glued down until the card was opened. It would appear the flap would be brought around to the front and cover up the previous bottom part of the card, which would also unfold downward along the center fold.
The Fold-A-Roos gimmick essentially reversed Mad magazine's Fold-Ins, drawn by the inimitable Al Jaffee. They measure about 2 1/2" x 4 11/16" in its closed state and the height increases to 3 9/16" once unfurled, according to Benjamin. They must not have tested well and may have been a bit hokey, even by the standards of the day.