Regarding the US national anthem, the story basically goes as this: during the seventh inning stretch of Game 1 of the 1918 World Series between the Boston Red Sox and the Chicago Cubs, the military band present at the game played the Star-Spangled Banner. It "stirred the crowd" and was one of the high moments of the game for the Cubs who were the heavy favorites, but were getting destroyed by Babe Ruth and the Red Sox. Harry Frazee, the owner of the Red Sox (who later would end up selling Babe Ruth) decided to play the song before every game when the series went to Boston.

The sense that this was the genesis of the tradition of playing it before all games is noted by both Marc Ferris in Star-Spangled Banner: The Unlikely Story of America's National Anthem as well as Geoffrey C. Ward in Baseball: An Illustrated History.

https://books.google.com/books?id=IEpjBAAAQBAJ&pg=PA133&lpg=PA133&dq#v=onepage&q=1918%20world%20series&f=false

https://books.google.com/books?id=_076VanTBE8C&pg=PA132&lpg=PA132&dq#v=onepage&q=patriotic%20fervor&f=false

And while I can't vouch for every fact, this article by ESPN is also quite decent for a more casual read: http://espn.go.com/espn/story/_/id/6957582/the-history-national-anthem-sports-espn-magazine

I cannot remark on how it spread to other sports, or if other nations do similar, but at least among baseball, this was the start.

I'm certainly not an expert on this topic, but one of my colleagues is, and he recently wrote about this on a blog he runs:

Patriot Games: Military Displays at Professional Football Games

Check it out!

This article says Wales responded to the New Zealand's traditional Haka by signing "Mae Hen Wlad Fy Nhadau" in 1905. It only mentions starting the tradition in games of Rugby but I can't find a mention of it happening in any other sporting event before this.