How is Newton's third law working here? I understand that if I were, for example, to throw a bowling ball in outer space then the ball would move away from me by the force I generated. But the ball would also exert a force on me in the opposite direction and move me away in the opposite direction of the bowling ball.
I don't understand how Newton's third law works in the next scenario though: I saw on YouTube an MIT professor let the gas out of a fire extinguisher on the back of his bicycle which then propelled him forward on the bicycle. What's exactly causing that motion? How is Newton's third law working here?
The gas molecules are pushing on neighboring gas molecules which is ultimately causing gas to escape from the nozzle. The gas molecules are pushing on one another like my bowling ball in space example, right? But where is the gas pushing on the fire extinguisher? Is there more pressure on one side of the fire extinguisher?
 A: Think about pressure in a cylinder containing gas at high pressure. If the cylinder is closed then the pressure is acting everywhere on all the walls of the cylinder. Now open one end of the cylinder. The gas at that end rushes out (and does not push on the cylinder). The gas at the other end is still pushing on the cylinder with a force $pA$ where $p$ is the pressure in the gas, and $A$ is the area of the end wall. As the gas escapes $p$ falls, but it can be quite high to begin with.
Similar things go on inside a rocket motor. The main difference is that fuel is fed in and burned so as to maintain the pressure.
Comment on Newton's third law
I would say it is a little unclear to appeal to Newton's third law in the case of a rocket. It is better to invoke conservation of momentum, but these two ideas are closely related. The third law is acting here at the collisions between gas molecules and wall of cylinder, and at the collisions among the gas molecules. The conservation of momentum is also acting in all those processes, and it can be invoked for the overall situation: the backwards momentum of the escaping gas is exactly balanced by the forwards momentum of the rocket.
Similar remarks apply to the recoil of a gun. The bullet does not cause the recoil. Rather, the explosion of the gunpower produces a large force on both the bullet and the gun. It is this force that causes both the acceleration of the bullet and the recoil of the gun. However, from the fact that a bullet is speeding away you can deduce that the gun must be recoiling.
The reasoning is like detective work. Sometimes you will see puddles of water forming on the ground, and at the same time people are putting on their coats and opening their umbrellas. Did the coats and umbrellas cause the puddles then? No. Rather, it has started raining and this caused both the puddles and the umbrellas. Similarly, the bullet does not cause the recoil. And for the rocket it is not the exhaust gas but the high-pressure gas inside the rocket which is causing the rocket to accelerate at any given time, by exerting a force on it.
A: When the gas is inside a container the molecules (red) are hitting all sides and exerting a force in all directions (due to Newton's 3rd law), first diagram.

When gas is allowed to escape, second diagram, the molecules are still hitting the wall at B and exerting a force, but no longer hitting the wall at A.  This causes an overall force to the right on the container.
A: The walls of the chamber that houses the compressed gas is pushed upon by the molecules inside the chamber. The actual gas molecules escaping through the nozzle do not push on the walls of the extinguisher though, but since there is pressure throughout the chamber, the escaping molecules near the nozzle push on the molecules inside the container that also push on those that are near, and at, the walls of the chamber.
In terms of Newton's third law, it may be more intuitive to think about it in terms of momentum conservation. If you have mass exiting in one direction from an object, then the object must move in the other direction with opposite momentum or $$m_gv_g+m_fv_f=0 \\
\rightarrow m_gv_g=-m_fv_f \\
p_g=-p_f$$
where $m_g$, $v_g$ is the mass and velocity of the gas, and $m_f$,$v_f$ is the mass and velocity of the fire extinguisher.
A: The gas molecules next to the metal of the fire extinguisher push on it. When these molecuoes bounce off the metal surface, the rest of the gas molecules in the container push them back towards the metal, keeping the pressure on.
A: This question can be easily understood if you apply law of momentum conservation. Now where is Newton's third law involved here? IMO, law of conservation of momentum is derived from newton's first law. For n particles, you calculate the velocity of the the centre of mass of the 'n' particles and if there is no external force acting on these particles then initial and final velocity of the centre of mass of the system remains constant which is essentially what Newton's 1st law states.
$$m_1u_1 + m_2u_2 +.......m_nu_n = m_1v_1 + m_2v_2 + ........m_nv_n$$
Note that Newton's third law also plays a role here , as all the internal forces cancel each other as a consequence of Newton's third law and dont need to accounted in the final equation for the velocity of the centre of mass
Now coming to Newton's third law.
An internal force causes some of the gas particles to gain velocity in say, a particular direction .This internal force is due to the haphazard motion of gas particles(caused due to the pressure difference, laws of diffusion etc etc...).According to newton's third law if one particle(B) pushes another gas particle A in east direction, then particle B will push particle A in west direction with force of equal magnitudes, and hence the opposite velocities.
