In general, noise is an error which is superimposed on top of a useful signal. Noise may be random (aka correlated or partially correlated) or systematic. Usually, noise is classified by type (additive, multiplicative, white, black, red, brownian, Gaussian, Cauchy, etc… each with its own mathematical description) and by the source origin (Johnson, cosmic, 1/f, quantum, Barkhausen, EM, etc… etc... etc… each being of a certain type).
Audio noise is any unwanted perturbation that comes on top of the useful signal. These perturbations can be of any type and any source, the only specific property is their limited bandwidth, to audio frequencies, so many high frequency noise sources can be safely ignored.
Sorry, your question is too fuzzy to further expand. Short answers, audio noise is not necessary random, the distribution of frequency and amplitude is not necessary relevant, and correlation (or not) with the signal is not specific to audio noise. Example, hum is a highly correlated (aka systematic) noise source of EM origin, tape hiss is a mixture of Barkhausen, 1/f, a few other sources, and is little correlated (aka random).