Those cards were probably ten years old. THey look a lot like some of the cards I am talking about. Once such things became the rage there were knockoffs from genuine Big Daddy Roth cards.
Playing hardball with 5 total guys was right up my alley. With five you could play Over The Line with everybody cycling through the positions of pitcher, infielder outfielder, batter, and a catcher. Did it many, many times at local schools and had a ball. It was best with 6 total, three per side,
Re: the TV shows, Gilligan's Island started in 1964 and overlapped the original Star Trek by about a year. Dobie Gillis started in 1959 and ended after 1963. Since reruns were more scarce back then, perhaps they just missed your youthful TV-watching days and neither was airing when you were young (?).
I'll never forget the first times I heard Elton John's first hit, "Your Song" as I walked through the outdoor corridors of my high school in late 1968. Despite the incredible variety of sounds that comprised late '60's music, we had never heard anything like Elton John.
The BIg Red Machine = The Archenemy. 'Nuff said. But if they had been MY team, I'd have enjoyed the ride through the 1970's.
(Disclaimer, I had to look up some of these dates. My memory ain't THAT good.)