If I push my hands against each other, why does the stronger one move the weaker one if the forces should be equal and opposite If I were pushing my hands against each other, the stronger arm will move the weaker arm, but if all forces have an equal and opposite reaction, why does the weaker one move?
 A: The force balance is real - but you may have to take into account acceleration of mass along the way.
Acceleration gives rise to an additional force - if I push onto a 1 kg mass with a force of 10 N, it will accelerate at 10 m/s/s. But if I put a second mass in front of the first, their total mass is 2 kg and when I push with a force of 10 N, they will accelerate with 5 m/s/s. If we look at the forces in the system we see:

Somehow, the left hand object "absorbed" 5 N of force, with 10 N on the left and 5 N on the right. This is because the object is accelerating. At every point there is balance. On the left, the 10 N from my hand is feeling a 10 N push-back (reaction force) from the first block; in the middle, the blocks have an action-reaction pair of 5 N. There is no contradiction here.
In the case of your hands pushing, it's the same thing. If one arm is pushing harder than the other, your hands will move in the direction of the weaker arm; at every point in the chain forces are balanced, but where there is accelerating mass there will be a change in force along the chain.
A: This is due to an imbalance in the forces exerted by your right and your left hand. While it is true that together they feel the same force, the force that is pressing them together, both your hands feel a net force as one. 
Similarly, you can push a box in one direction, even though, the reaction forces between you and the box are equal. 
The next statement I will make is not entirely correct but it may help you picture what is going on. When you push the box, you may be able to move it even if you are lighter than it. In a sense you have more inertia (not entirely true it is not really inertia) because you are exerting another force that allows you to be more rigid than the box.
The same goes for the scenario when you press your hands together. Through some other force exerted by your stronger arm it is more inert and less reactive to the other arms force.
If you don't understand I can try to draw a diagram.
A: Because your system does not only contain your hands, if the weaker arm is moved then you most probable experience a rotation with your upper body. Additionally you are also connected to the ground, which means some of the force will be hand off to the earth.
