A certain famous Disney star appears in one shot of the film. Momentarily as we see the Mars Recovery vehicle approach the planet, Mars itself, the rotating circular hub of the spaceship, and the round satellite dish sitting on its spine line up to form the friendly and familiar Mickey Mouse logo.
There were theme park attractions at Disneyland and the Magic Kingdom at Walt Disney World called Mission to Mars. Both opened in 1975, but Disneyland's closed in November 1992, while the Magic Kingdom's ran until October of the next year.
Many of the props from this movie, including the big turning wheel set and the spacecraft model are now on display in the line queue area at the "Mission Space" ride at Epcot in Walt Disney World in Florida.
Gary Sinise's fancy convertible in the beginning of the barbecue scene is an Isuzu Vehicross convertible, a halo car produced for a very short time from '97-2001, selling only 4,153 units in the U.S.. It was never introduced as a convertible in the US. The jet sounds were added to give it a futuristic engine effect.
After Mars Recovery is given the OK to enter orbit around Mars, Woody Blake says "Let's light this candle". This is exactly what Alan Shepard, the first American in space, said just before lift-off on his inaugural Mercury flight.
The "face" is apparently based on a photograph taken from the Viking orbiter in the 1970s, showing what appeared to be a face-like shape in the surface features. Later photographs from a more advanced probe showed the same feature at higher resolution and different lighting, and it looks nothing like a face.
The trivia items below may give away important plot points.
The rotating circles that signify the alien countdown are indeed *counting* down... in binary. The circles in their horizontal and vertical configurations correspond to zeroes (0) and ones (1) respectively, and the least significant bit (LSB) on the viewer's left. The sequence in which they turn correspond to a binary countdown. The leftmost lit circle is not part of the binary count. There is an initial pattern of "all horizontal", following which the countdown actually begins (starting with "all vertical").