Now, I'm not an engineer, but I've got a general understanding of physics... still, there might be some technical errors in here, so don't jump on me if these descriptions don't quite hold true to physics.
Quick cars
The Quick Cars were most likely powered by frictionless flywheels. Flywheels -- frictionless wheels that spin at great speeds, storing energy -- are capable of storing energy for a very extended period of time, even 500 years, if constructed properly. They lose very little energy over time, and are quiet. The quick cars were started by pushing a lever; this probably simply lowered the already spinning wheel onto the track, in the same way that a car, lifted in the air, could spin its wheels and not move anywhere until it was set on the ground.
The Gummiscope
This is just one of those two-lens focal point contraptions, except much bigger. There is one inner convex lens, the focal point of which is at the center of the outer, concave lens, the result being a concentrated beam of light. There are also some smaller lenses (or possibly mirrors) on the inside of the machine that always point towards the sun, and bend around the light to the desired direction.
The Gummarine
This is a very interesting contraption, and demonstrates some of the true hidden technologies of the Ancient Gummis. When Tummi opens up the bottom of the Gummarine to sabotage it, he finds a complex maze of pipes, and by rearranging them, the controls function imporperly. The controls of the Gummarine were most likely a vacuum tube computer, and many other Gummi devices may have had similar hidden vacuum tube computer systems -- The Gummarine's system was quite disguised from normal view.
The Flying Machine
Most likely, this is a helium balloon, not a hydrogen balloon (because the air wasn't heated first). The propulsion system is also of interest because it is powered by gummiberry juice, a sign that the juice probably contains some high-energy organic compounds.
Gummiberry Juice
This is by far the hardest to explain... The first half of the recipie deals with the ratio of one color to another, and by viewing results of failed or sabotaged recipies, it is obvious that the color ratio is very important. This leads me to suspect that the active ingredients in gummiberry juice are the dyes of the berries. The second half of the recipie is the famous three-step stir... the stirring action is probably to mix the juice with the air, and the juice reacts with something in the air, probably oxygen. This is a volatile chemical reaction, (which is where the slowly to the left, slowly to the right comes in) as many organic reactions are, and it also produces another gas by-product which must be removed by tapping the side of the bowl to get the bubbles out. Of course, this still leaves open what the juice actually does. For the 'correct' recipie, it probably stimulates adrenaline (the 'fight-or-flight' reaction, which evolved differently in humans and gummis). As for the incorrect recipies, I don't know how those begin to work.