Do propeller coefficients go to zero at the same value of advance ratio?
Do $C_P$, $C_T$, and $\eta$ always go to zero simultaneously at the
    same value of $J$ for all propellers?  Why?

Yes; Talking in dimensional quantities, This speed, at which all coefficients goes to zero is called pitch speed. 

At pitch speed thrust goes to zero due to the fact that the incoming airflow drives the apparent angle of attack seen by the blades to zero.

Power = Thrust * velocity , As thrust goes to zero power goes to zero as well.   

Efficiency $\eta = \frac{P_{out}}{P_{in}}$, $P_{out}$ goes to zero as shown above hence $\eta$ goes to zero.

HTH