Using a zener diode to limit current to an IC for various input voltages
1) Correct. $$R_S = \frac{V_{in}-15V} {1mA + I_{aux}} = \frac{24V-15V} {1mA + 1.5mA} = 3.6 k\Omega $$

2) At 48V, you should pick a different resistor. $$R_S = \frac{V_{in}-15V} {1mA + I_{aux}} = \frac{48V-15V} {1mA + 1.5mA} = 13.2 k\Omega $$  

3) Yes. It's quite weird. The \$19.4V/24V \approx 4/5\$.
But \$4/5*480\Omega = 384\Omega\$. And actually \$470\Omega\$ starts to make more sense.
$$\frac{20V-15V}{470\Omega}=10.6mA = 8mA \text{ (for }I_{aux}) + 2.6mA \text{ (for } I_Q)$$ But that requires a big R1 and D1 at higher input voltages.

BTW, I didn't find a hard upper current limit for \$I_Q\$ in the datasheet, only a recommended. Maybe 8mA is allowed and the 300 Ohm works? But maybe it hits the thermal limit when exceeding 3mA?

4) Don't forget to subtract \$I_{aux}\$, but 4.5mA is still more than recommended

EDIT: based on replies
Page 5 shows the recommended values for \$I_Q\$, not the absolute maximum ratings.

An approach to estimate the maximum current is reasoning as follows:
Were it a normal zener, the voltage would be clamped to 15V. For the case with the \$300\Omega\$ resistor, up to 8 mA flows through the zener. For a normal zener, this would yield a dissipation of 120mW. 120mW causes using a blunty approach a temperature rise of only \$R_{\theta JA} *120mW = 183.8 °C/W*120mW = 22°C \$.
This applies to a normal zener, but I think their implementation of the ideal zener will not much deviate from this dissipation (If it was worse, why did they not implement a real zener in the DRV110 in the first place??).

So, why the recommendation of max 3mA if it is not for limiting the power dissipation? Took some time, but I think it has to be explained as follows.
The footnote says:


  The device sinks up to 3 mA with the added supply current.  


The device uses up to 3mA at max, not the zener! So, picking a too large \$R_S\$ might drop the supply voltage below the 15V, so the zener is disabled/not functioning. Normal zeners require a current to clamp at the rated voltage (zener datasheets show a test current), which is probably the reason a higher current should be choosen than the sum of \$I_Q+I_{drv}+I_{aux}\$.  

But then, it's weird they use 1mA in their calculations, and not the max 3 mA.