Does layer order matter on multi layer PCB?
It depends on your design requirements, such as highest signal frequency and EMC (electro-magnetic compliance).  But in general for audio type signals it doesn't matter too much. The power plane will act like a ground plane next to signal layers, so you want to use as much copper as possible on the power planes as well. If your traces need to match a specific impedance, then you'll need to calculate trace size based on the dielectric and board thickness. There are good trace calculators on the web. I'll provide an example below. Or you will need a good layout CAD program which can computer trace impedance based trace dimensions and how far away the underlying ground/power plane is. 

For a simple audio circuit like you have you really probably don't need to worry about trace impedance too much assuming your trace lengths are far shorter than a 20 Khz or so wavelength (20 KHz has a wavelength of about 15 km!). 

Trace Impedance Example
 Using EEWeb Trace Calculator:


Board Type FR4: dielectric constant = 4.7
Layer thickness (between layers): 40 mil
Copper thickness:  1 mil 
Trace Width: 50 mil
Resulting Impedance: 61 ohms


Layer Stackup

I would definitely modify the layer stack up as follows:


Top connectors and signal traces
Power
Ground
Signal


The above is typically what is seen. I like to have the signal layers on the outside in case any ECs (engineering changes) are needed. It is easy to modify traces on the outside, but not inside.