The schematics you show tells how the grounds are connected at this board only, not how they are connected in the whole system.
So answer to your first question is, no, it can't work like that by itself. In this mixer, the system designer has chosen to use star grounding, where all the different grounds are only connected at one point only, on the power supply board, to avoid ground loops.
The capacitor is there to allow for AC signals (the audio) to have a short return path, so that they are not affected by wire inductance.
So the answer to your second question is, no, it is not a good idea to use a resistor, because current through resistor creates a voltage difference. In this mixer, the system design has chosen a 0 ohm resistor basically, as the wires have extremely small resistance so that small currents do not cause much of a voltage difference between different grounds.
The star grounding is the key to make a quiet system here, so for example the headphones output ground return wire is a separate wire to power supply board, so that for example, sensitive mic input ground wire does not carry the headphone ground return currents.
As to your final question, just run signal ground where you need it, and power ground where you need it. Audio signals are not that prone to couple between tracks, but you can use ground planes if you like. Some systems can work with single ground plane for both signal and power, but it still requires some careful thought how to route them, so that signal grounds do not carry power grounds and cause coupling that way. Basically, that answer would require a book about the subject, and there are books about grounding and audio devices.
Shield of a shielded cable such as RCA input is return path for the signal, so it belongs to the signal ground. The mixer uses signal ground for the XLR input as well.