Electrical – Analog Stick Wiring

You are not going to be able to distinguish potholes clearly from other short peak events apart from being able to distinguish between a rising bump in the road and a hole (the intial direction will be opposite) but you can certainly capture them quite easily.
Determine an initial direction (e.g. negative/positive XYZ depending on how your device is mounted), a threshold level, and a maximum time the reading should be over this level (determined by width of pothole) Then time the peak height/width and see if it fits your pothole characteristic.
The device already contains an internal 1kHz LPF, so you could add a HPF of say 50-200Hz for the potholes, since they will have a fast risetime. I'm not an expert on car vibration frequencies, but you will probably get some noise from vibration however you filter. However that's not an issue as long as the pot hole event is large in comparison with the noise - it looks like the data is okay as it is, I would just sample a bit faster to prevent aliasing (e.g. >2kHz) or add a LPF to the existing internal one as described in the datasheet. Since you are trying to capture fast risetime events, I'd go with the former (faster sampling, possibly with HPF)

To compensate for a change in inclination, you can have a running average value which can be used to zero the axis out (one for each axis). Also, note that a HPF will ignore the DC level, so (as long as it doesn't go off the end of the scale) a slow gradient will make no difference.

According to the datasheet (bottom of page 7 in the link above), the formula for the external capacitance is:

\$ C2 = C3 = C4 = \dfrac{4.97 \times 10^{-6}}{f_{BW}} \$

so your calculation of:

\$ \dfrac{4.97 \times 10^{-6}}{10Hz} = 497nF \$ is correct.

1. Physically disconnect all connections of the module from the Arduino.
2. Then, check the resistance between the Vcc pin of the module and the actual pins at each end of each little potentiometer in the joystick device on the module.
3. Similarly, check resistance between the Ground pin of the module and the physical pins at each end of each potentiometer in the joystick device.
4. Finally check the resistance between the VRx pin and each of the center contacts of the little potentiometers, and similarly with VRy.

In a working module, in each case you should see a zero or nearly zero (~10 Ohms) resistance for one specific potentiometer pin from the module pin.

Almost definitely the soldering of one of the device pins on the module board is not perfect - Most likely the Vcc pin to one of the X-axis potentiometer's pins won't give you the low resistance reading expected.

Re-solder that connection on the little PCB.