What would happen if every person in the world gathered at a minimum possible area and jumped all at once? Would that have any effect on Earth?
First, obligatory note: this was already answered by Randall Munroe of XKCD fame in his "What if blog" (this was his 8th question & answer posted).
The answer, as given by Randall (emphasis mine)
it doesn’t really affect the planet. Earth outweighs us by a factor of over ten trillion. On average, we humans can vertically jump maybe half a meter on a good day. Even if the Earth were rigid and responded instantly, it would be pushed down by less than an atom’s width
The problem is the 7 billion-ish people leaving said area.
Not much. Very little, and only for a short while.
Let's say all those people can jump about a foot high, or 300 mm. Note that this is the total center of mass height change while in flight, not how high the bottom of your feet get from the ground. At 1g, a ballistic flight of 300 mm height takes 495 ms, or 1/2 second.
At the beginning of that 1/2 second, there will be a impulse that starts the earth moving ever so slightly in the down direction at the location of the jumpers. Meanwhile the gravity force on the jumpers will accellerate the earth upwards. The effect of the initial impulse is canceled at the peak of the jump. During the second half of the jump, the gravity force will cause the earth to accellerate upwards. When everyone hits the ground again, the earth experiences another impulse that exactly cancels everything out, and everything is back in the same position and at the same velocity it started out at. No net displacement or change in velocity will have occurred.
Now let's look at the energy. Let's say the average person has a mass of 60 kg, so a weight of 600 N. 600 N raised 300 mm is 180 J. Times 7G is 1.3 TJ. To put that in bomb perspective, a kTon of TNT is considered to be 4.2 TJ, so all that jumping does't even add up to a third of a kiloton TNT equivalent. Yawn.
Just for fun, how small a area can these people be packed into? Let's say to "jump" reasonably, everyone needs a square meter, so we need 7 Gm land area, or 2700 square miles. That would be a circle 47 km in radius, or 94 km in diameter, or 59 miles in diameter. That is a area about 25% larger than Rhode Island, or about half the area of Connecticut.
So nothing of consequence would happen to the earth, but there would be one heck of a traffic jam getting people in and out, all the porta-potties, the save the earth protestors demonstrating about everything getting trampled, the nutcases predicting massive earthquakes and floods, scalpers selling portable chairs for greatly inflated prices, etc, etc. What a mess. Please do it on the other side of the earth from wherever I happen to be at.