Choosing Component Values for DCO Design
My question is, which value of C1, RS2 or max current is chosen first, and why?


Generally we'll pick the capacitor first. They come in fewer values per decade (E3 or E6 series) than resistors which are available in E12 or E24 series. Pick the capacitor and then calculate the resistor based on that. But the two are intertwined ...

We want to pick a mid-range resistor value. Too low and we'll load the previous stage and need a large capacitor. Too high and we'll suffer bias-current offsets and integration of the bias current. 10k might be a nice value. It would limit the current to 0.5 mA on a 5 V input.

From \$ Q = CV \$ we can differentiate and get \$ I = \frac{dQ}{dt} = C \frac {dV}{dt} \$. Rearranging this we get \$ C = \frac {I}{\frac {dV}{dt}} \$. Using your values of 5 V / 500 µs = 10000 V/s we get \$ C = \frac {0.5m}{10k} = 0.05 \ \text {µF} \$. 

Let's increase this to 0.1 µF for convenience and recalculate. \$ I = C \frac {dV}{dt} = 0.1µ \times 10000 = 1 \ \text {mA} \$.

For 1 mA from a 5 V supply we can calculate \$ R_{S2} = \frac {V}{I} = \frac {5}{1m} = 5 \ \mathrm {k\Omega }\$.