How should PCB traces be treated in different situations?(as capacitance, resistance or inductance)
I know that for signal integrity, the PCB traces are always considered to be part of the electrical circuit.

Not always. Slow edge digital signals, non-precision low current analogue circuits, will usually get away with anything.

How should traces be considered in different situations, like high current, RF, high edge rates etc.?

For high current, the resistance of the track is usually significant. Make sure it's wide enough.
For RF and high edge rates, the length is significant if it is longer than fraction of the wavelength of the signal or edge. Then some form of controlled impedance for the line, and matching to it, is needed at one or both ends of the track, depending on how the line is being used.
In precision analogue, you need to make sure a current through the resistance or inductance of a trace doesn't generate a voltage that's impressed on some other part of the circuit that it should not be. This is often easier said than done.