It sounds like you are talking about Archibald "Moonlight" Graham, who played one game in the major leagues and never got to the plate. He is featured in the movie "Field of Dreams" and also in the book on which that movie was based, " Shoeless Joe " by W.P. Kinsella. In fact this has happened to many, many people. The shortest career I know of was that of Larry Yount, Robin Yount's older brother, who was a pitcher, and was called into his first game as a reliever, had his name announced so he was officially the pitcher of record, then hurt his arm doing his warm-up throws from the mound, had to come out, and never played again.
Steve Finnan is the youngest player to have played in all top 4 English Leagues at the age of 19
Certainly. Too many to name
Professional? Yes -- Toni Stone played in the Negro Leagues in the 1950s. She also played in the Negro minor leagues in the 1940s.
The answer to that question is: Christian Poulsen