Audio Spectrum Link to Coposition (with music inside)

From your description, the easiest thing to do would be to pre-compose the solid and the audio track you want to use, then apply Audio Spectrum to the pre-comp. That way Audio Spectrum would be using the same layer for the reference audio. After you have the effect set up the way you want it just press E to reveal all effects on the layer, draw a selection around the effects and then go to Animate>Save Animation Preset. Put it in the default folder for custom presets and it will show up in your Effects and Presets panel. You can also save the folder to DropBox and then create an alias to that folder that you place in the default location for user presets and it will show up on every machine you work on. That is what I do.

Now anytime you want to apply that template to any comp all you have to do is add a solid, add the audio track, pre-compose and apply the animation preset you saved. If it were me every new song would at least be a separate comp. I have created over 200 of my own custom animation presets for everything to automating animation to color corrections and I use some of them on almost every comp I create. They are huge time savers.

As for saving new comps for each new project and organizing your files my best advice is to carefully develop and religiously follow a carefully thought out file structure. Mine may not work for you but it was developed way before we used computers in post-production and is based on the old movie industry standard way of keeping track of projects, scripts, scenes, shots, and clips. I can find anything I have ever done using that system in a few minutes. More than 90% of the people I work with have lousy file management skills so don't feel too bad if you have not developed one yet, but once you do it will be much easier to get home in time for dinner.