The difference between the "mouth hole" and the "needle hole" is that the mouth hole is reinforced and under low tension, while the needle hole is not reinforced and under high tension.
The mouth piece of a balloon, some might call it the "neck" of the balloon, generally includes a rolled section of material. This section is what I call it being "reinforced," since it is stronger than the rest of the balloon. Additionally, the neck is under low tension compared to the stretched fabric of the rest of the balloon.
So, why should any of that matter?
The tension on the surface of the balloon determines the response the balloon material will provide under a given load. When you poke a hole in, or open, the neck, not very much happens, because there is low tension (low force). However, when you poke a hole in the body of the balloon, the much higher tension there provides a larger response.
You can make an analogy to a stretched rubber band. If you stretch it a small amount, it is easy to perturb the rubber band a great distance. However, if the rubber band is stretched to near breaking, then even a small perturbation results in a snapping of the material, and the subsequent release of energy (both in motion of the pieces and in generation of sound).
It travels forwards instead of backwards in an accelerating car for the same reason that a helium balloon travels upwards instead of downwards under the influence of gravity. Why is that?
In an accelerating car, for all intents and purposes the acceleration can be considered a change in the amount and direction of gravity, from pointing straight down to pointing downwards and backwards. The balloon doesn't know and doesn't care if the acceleration is from gravity or from the acceleration of the car; it just tries to move in the direction it naturally moves, namely, against the direction of the acceleration. Thus, it moves forwards when you accelerate. Hopefully you find this explanation intuitively satisfying.
Another more rigorous way to view the problem is through Lagrangian minimization. The balloon can be considered a low-density object embedded in a higher-density fluid constrained within the confines of the car. Under the influence of gravity pointing sideways, the total system potential energy decreases the farther forward the balloon is situated. Since the force is the gradient of the potential, the balloon will try to move forward.
Best Answer
A link to that experiment will be welcome, but even without it :
If you push the balloon against a needle, this single needle will have to resist to your push on a very small area of the balloon, (its tip, essentially) thus exerting a very high pressure that would burst the skin of the balloon before the general shape of the balloon changes enough to adapt.
If you push it against many needles, the push will be shared and the pressure on each tip will be divided by the number of needles. And if you try to push harder, it will not work so easily, because then the whole balloon will change shape and spread in directions where there are no needles, without exerting much more push on the general area where the set of needles is concentrated.