How did the players decide what to play? Were there set pieces that they improvised on as the scenes demanded (sad, rousing, suspense, ...)? Did they watch the movies a couple of times beforehand or did they jump right in? Were there training schools to become a movie accompanist?
Also: did some movies have official music scores?
All sorts of audio accompaniment took place at silent film showings, depending on the film, period, and locale. Silent film spans nearly 30 years and the world over, so there's really no single answer. Musical accompaniment could be live or pre-recorded (Thanks to, for example, a gramophone). It could be a pianist or a musical ensemble; many silent era movie palaces have a full orchestra pit that would be used to give silent films a complete orchestral score, live, as well as being used for other kinds of performance besides cinema.
It could even be absent. An alternative to musical accompaniment was a live narrator - In Japan, a narrator (Benshi) would contextualise the story and even voice the characters on screen. So there really is no single standard and I assume all possible approaches were taken at one point or another. Movie theatres in the USA originated as an outgrowth of vaudeville and other popular entertainment, which of course means there was already a cadre of working musicians to fill that role.
Some silent films did have "official" scores - Chaplin notably composed scores for several of his films, including silent ones, and those scores would accompany the film itself on distribution along with strict (And of course, oft-ignored) instructions to play Chaplin's score and only his score. The various restored versions of Chaplin's films that you can find on streaming and home video versions were mostly put together by recording versions of the original, restored scores with help from Chaplin and his estate.