Telegrams from Western Union were the way that the armed forces notified the wives and parents of soldiers killed during WWII. As such, everyone knew what the telegram was likely to be about, and all of the married women were worried it was going to be about their husband. When the Western Union guy started to walk away citing an error in his list, Jimmy grabbed it from the delivery guy so he could identify who it was for, in this case it was Betty. If Jimmy'd let the guy leave, every married woman on the team wouldn't have played their best and they would've likely lost the game. In this case, the WU guy was also acting quite insensitively about the situation and Jimmy was probably offended by his behavior.

It was also a way to show Jimmy's sensitive side after he'd been established as a pretty hard-nosed coach and that the war was affecting everyone, including Dottie who hadn't heard from her husband in weeks and was worried about him.

Standings were not shown during movie, so we can think that World Series were played between #1 and #2 team. Though actual AAGPBL implied the rule that World Series should be played between champions of both halves of the season.