what is the purpose of the ground and the capacitor in this circuit?

Why grounded? the alternative is to let if float, and that creates two problems, first, the amplifier may not handle floating (differential) inputs, and second, you will get noise pickup from the EM fields around the wires. I'm assuming the cable is shielded.

If your amp has a differential input, you may be able to not ground the wire, but you still need a shield around both wires, and that has to be grounded. In general, you should follow the manufacturer's recommendations.

b) The purpose, if it has one, depends on the pickup characteristics and specifications. It may be specified by the pickup manufacturer to shape the response curve, or reduce high frequency noise. Again, you should follow the manufacturer's recommendations.

Typically, the wiring between guitar and amplifier would be shielded cable and the ground probably represents the shield ground. If the shield is not grounded at one point (and only one point), the wiring between guitar and amplifer/speaker could act as an antenna and other signals, such as radio signals, could be introduced to the input of the amplifier. Without the ground, you may hear the audio of close by police radio transmissions along with your music.

The RC network between the guitar pickup output terminals seems to be a high pass filter the pass through frequency would be adjustable by the potentiometer. Since no values of the capacitor or resistors are given, I couldn't speculate on the purpose of the filter.