Anderson's now-signature tracking shot style came about because of bad weather.
The aquarium groundbreaking scene in Rushmore was initially meant to be more straightforward, but after a huge rainstorm the day before shooting turned the baseball diamond location into mud, Anderson chose to keep the focus on the foreground action rather than the nasty background. He liked the result so much, he's recreated it in nearly every film since.
Bill Murray paid out of his own pocket for the Rushmore helicopter scene.
Disney wouldn't shell out the $25,000 for the chopper rental to get a shot Anderson wanted for part of a montage, so Murray cut him a personal check for the amount. The scene never ended up in the final cut of the montage (which you can see here), and Anderson never actually cashed the check, which he's held onto to this day.