The film isn’t pulled through the camera or projector continuously, the movement is intermittent. It moves forward one frame, stops, then moves again.
In both the camera and projector there’s a shutter, but the shutters are different. In the camera, there’s a disk with a wedge cut out of it (how big this wedge is depends on the camera and in some cameras can be variable) that spins continuously. When the film is moving, the solid part is in front of the lens and blocks the light, so there is no motion blur (at least from the film moving, there can be motion blur if the object being filmed is moving too fast). When the film is stationary, the open part is in front of the lens, so the film is exposed.
There’s also another style of shutter, where instead of a spinning disk with a wedge taken out there’s a guillotine that drops and raises, but the purpose is the same.
There are shutters in all camera except those designed for very high-speed filming, where trying to move the film intermittently would just succeed in shredding it to bits. There, the film does move continuously and, instead of a shutter, there’s a rotating mirrored device that tracks the picture onto the film as it moves by.
In the projector, the shutter still has a solid portion that covers the film as it’s moving, so you don’t see a vertical blur (although you wouldn’t see much of one anyway, the intermittentness of the picture is enough to fool your brain, as is evident from many of the old shutterless toy projectors of the '20s), but it also has one to four additional solid segments that block the film even while it’s not moving. These are to minimize the flicker that would otherwise be extremely evident, at the expense of picture brightness (necessitating extremely powerful light sources for projection).
24fps is what you’ll see in a cinema, with very few exceptions, but outside of one, all bets are off. You can film at damn near any speed you want, although anything below around 12fps will be distractingly jerky and have very blurred motion. 9,12, 16, 18, 24, 25, 36, and 54fps are common settings on home cameras like Super 8.
As to processing exposed film, you have professional machines like have been described that pull film steadily though various baths, but then you also have simpler devices that can be used at home or for other small scale stuff or when processing is needed in the field. For smaller gauges like 8, 9.5, and 16mm, you can use spiral tanks, which are exactly the same as the ones you’d use to develop still film, but much larger. For something like 35mm I don’t know if a spiral has ever been made for it (it would be mammoth), but you can develop it in a rewind tank. In one of these, there’s a tank with two reels in it. You wind the film onto one reel and connect the end to the other. Then, you add you chemicals to the tank one at a time while continuously winding the film from one side of the tank to the other. And, of course, there’s always the low-tech bucket method (step one: bunch the film up into a big tangled ball; step two: put the ball in a bucket and pour chemicals over it).