I understand that when a ball collides with a wall it exerts a force. By Newton's 3rd law, the wall exerts an equal and opposite force back on the ball. As a result in an ideal world, the ball will bounce back at the same speed as before, conserving momentum. Is there a mistake in this intuition?
Unfortunately yes. You are not thinking clearly about system, internal forces and external forces.
Momentum is conserved when external force on system is zero.
What is the system here. There are two options.
If you consider ball alone, then momentum is not conserved. The ball experiences an external force during collision. Momentum conservation is not valid. In any case, we can see the momentum has changed. The direction of momentum has changed.
The other option is to consider ball + wall as system. Here wall is connected to rest of building. So again momentum conservation is not valid. Even if the wall was lying on a friction-less surface, huge mass of wall would make applying momentum conservation impractical.
In addition, I don't understand how the ball can travel back at the same speed as before when hitting the wall. Wouldn't the wall have to exert twice the force ( 2 times the force the ball exerts on the wall ) to do this?...
Here you are mixing up force, impulse, momentum and speed. You are thinking about mechanism of ball stopping and returning correctly.
Impulse is change in momentum. Force is impulse / time.
...The way I understand this, is when the wall exerts a force equal in magnitude to the force exerted by the ball on the wall, it should bring the ball to halt. ...
Correct idea with wrong terms. When ball stops, wall applies impulse equal to momentum of ball.
...It should then exert this same force again to send the ball back in the opposite direction. So shouldn't the wall exert twice the force?
Not same force, same impulse! So net impulse by wall would be twice the momentum of ball.
Of course this double impulse will act on wall also (as per Newton's third Law). Hopefully our wall is strong enough to handle that.