I found this essay by the Smithsonian magazine's Natasha Geiling that answered many of my questions.
One of the most interesting things in the article is how the most commonly used Olympics themes in broadcast coverage, "Bugler's Dream" by Leo Arnaud, wasn't written for the Olympics. How did it get associated with the Olympics, then? Well, someone at ABC decided to use it with their coverage of the 1968 Winter Olympics. And it stuck. To me, this music just screams Olympics, but sporting events were not on the composer's mind when he wrote it. Also interesting is that this preference is entirely American broadcast-based—people outside the U.S. don't use the same music at all.
Also used extensively in NBC's coverage is John William's "Olympic Fanfare and Theme" (see above—and why don't the Olympics use marching bands more?). William's theme was written for the 1984 Los Angeles Summer Olympics, and was also adopted for Olympic broadcast coverage, especially the summer Olympics. But, interestingly enough, the theme NBC most often uses in their coverage is a hybrid, starting out with Arnaud's and then splicing into Williams'.
Besides television coverage, however, neither theme shows up during the actually games—not in the opening and closing ceremonies nor anywhere else—unlike the Olympic Hymn, which is official. But I'm okay with NBC's music, because I think the orchestral brass fanfares add gravity to Winter Olympics events that otherwise might fall short. For instance, slopestyle. Did anyone see that Swedish skier with the pants that were way to big? And carried a raw egg in his pocket? The egg broke, if you were wondering.
What's your favorite Olympic theme?
Vocab: orchestral, brass