I was introduced to this contest at Buffalo State's summer physics teacher's institute (I highly recommend the program to anyone entering physics teaching or who has only taught for a few years)
We do several events throughout the year now. One of the standards is engineering design. Many teachers ignore it because it's not assessed on the state exam. I no longer ignore it... it's another thing that makes the course more fun.
Other events I've done:
Stack a certain number of bricks - only 1 brick can be the base (no other bricks touching the floor.) Group whose stack extends the farthest horizontally wins.
Construct an airplane from tape, straws, and tissue paper. (this event was a flop - or rather, every single plane flopped. I won't do it again in the future)
From ONE sheet of regular copy paper, build a tower. Tallest tower wins. Materials: 1 sheet of paper, a couple pairs of scissors, about 2 feet of tape. Record tower in 30 minutes was just over 2 meters. I've seen much taller from a single sheet of paper.
Slight deviation on your event #3, the egg drop:
We start at 1 meter, move to 2 meters, then 3 meters, then 4 meters, then 5 meters, 7.5 meters, 10 meters... up to the limit we have at our school. The egg is not surrounded by a device, but rather, the built device has to catch the egg without breaking it. I've seen the contest scored by the quotient of the height dropped and the height of the device. To make it simpler, I just limit the height of the catching device to about a foot. Materials are somewhat random, but usually some string, tape, a few pieces of paper (including the instructions), straws, manila folder, a styrofoam cup or two, and maybe a few other things that I have laying around.
At the lower heights, there is one glaring engineering problem that is often overlooked: the egg survives the impact with the device, but then it bounces up, out of the device, and onto the floor where it cracks. It doesn't have to go up too far to break when it hits the floor - 6 inches is sufficient - so, even rolling off the top of the device sometimes results in a cracked egg.
I'm proud to say I built a device (out of a manila folder, a couple pieces of string, a styrofoam plate and cup, and masking tape) which will catch an egg without breaking it from an unknown height. It survived 4 or 5 stories with no problem, but we missed the device when we dropped it down the stairwell from the top floor (6th floor?) Since aim isn't supposed to be a factor, a plumb bob is suspended next to the "drop zone" for everyone to use in lining up their drop. Unfortunately, the plumb bob was touching something which pushed it out of alignment on our final drop. I calculated, based on the way the device worked, that the device would successfully save an egg dropped from about 12-13 stories (as long as the egg hits the device.) If the egg reaches terminal velocity at that height, then the height is pretty much unlimited (again, as long as the egg hits the device.)
Tomorrow in class, and the next few lab periods, my class is starting their final competition: build a rocket (powered by a 2 liter bottle filled partially with water and compressed air to 100 psi) that will launch an egg into the air. The egg that's in the air the longest, without breaking, wins. Actually, I calculate the winner this way: time in the air, multiplied by 2 if the egg doesn't break. Thus, if everyone's egg breaks, the last group can't win simply by having their rocket tip over. (t = 0.01 seconds) Then, if someone figures out how to get the rocket to go up 5 thousand feet (actually, that height probably isn't possible - conservation of energy), then they deserve to win, even if their egg breaks. Winning times are around 30-40 seconds. It's damn near impossible for them to design parachutes that work effectively. (or rather, they overlook how critical this aspect is) Parachutes - maybe they make a perfect parachute, but it's TWO parts that matter: 1. The parachute descends very slowly with the egg in some sort of capsule. 2. (more important) The parachute actually deploys. Model rockets: an extra charge that goes bang, ejecting the parachute after a certain period of time. These rockets: no explosive devices allowed. Deployment of parachute is quite a challenge.