A boy presses his palms against the palms of another girl on an ice rink. They exert an equal force on each other and move in opposite directions.
This may seem quite silly but what about a large object resting on a surface. The object applies a force (mg) on the surface and the surface applies an equal force on the object. It would seem like the object should move up even with gravity present. The two forces in this scenario are applied to two bodies so Newton's Third Law should apply.
You are right, Newton’s third law does apply in that scenario, and the surface does apply an upwards force called the normal force.
Newton’s second law also applies, which says $\Sigma F=ma$ where $\Sigma F$ is the sum of all forces on the object called the net force. In this case there is another force acting on the object, the weight. If the object is not accelerating then those two forces must sum to zero net force.
An object accelerates when the net force (i.e., the sum of all forces) acting on it is non zero.
The force between objects that push on each other is called normal force. The normal force acting between the two skaters pushes each away from the other, and the reason why they move away from each other is, there's nothing to stop them. The boy feels the normal force pushing him to the west, and he feels nothing pushing him to the east, so he moves west. The girl feels the normal force pushing her to the east, but there's nothing to stop her from going in that direction, so off she goes.
The skaters fly apart because each skater experiences a non-zero net force.
The normal force between the skaters is caused by the action of their muscles causing stress and strain within their limbs and, by the fact that their hands can not physically pass through each other.
Two forces act between the rock and the Earth: Gravity is described in Newtonian kinematics as a force, which acts at a distance, attracting the rock and the Earth toward each other. The normal force is caused by gravity and by the fact that they rock and the Earth can not physically pass through each other. It exactly equals the force of gravity, but it acts in the opposite direction: It balances the gravitational force.
The net forces acting on the rock and acting the Earth are zero, so the rock and the Earth do not accelerate in any direction relative to each other.
You can see this by drawing free body diagrams for each situation.
Take the boy and girl. Let’s say the girl is on the left. Do a free body diagram on the boy. You have to replace the girl with the force she is exerting on the boy to the right. If we assume zero friction on the ice, there is no other external force acting on the boy, so there is a net force acting to the right and he accelerates.
Now let’s take the large object resting on the ice. Do a free body on the object by removing the ice surface. You need to replace the surface with the force the surface exerts on the object, namely $+mg$. But there is also gravity exerting a force $-mg$ on the object. The net force on the object is therefore zero. It does not accelerate up.
Hope this helps.
