Do propeller coefficients go to zero at the same value of advance ratio? Graphs of power coefficient ($C_P$), thrust coefficient ($C_T$), and efficiency ($\eta=JC_T/C_P$) of the 5868-R6 propeller as functions of advance ratio ($J=V/(nD)$) and propeller pitch (blade angle at 0.75R?) are found in many textbooks (such as McCormic): 







From these curves it seems that for a given propeller pitch $C_P$, $C_T$, and $\eta$ go to zero at the same value of $J$. At also seems that the slopes of the $C_T$ vs $J$ curves for higher value of $J$ (where $C_T$ approaches zero) are the same.

I have two questions:


Do $C_P$, $C_T$, and $\eta$ always go to zero simultaneously at the
same value of $J$ for all propellers?  Why?
Are the slopes of the $C_T$ vs $J$ curves where
$C_T\rightarrow 0$ the same for all values of propeller pitch?
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