I think that the above desktop suggestion might give you a problem with location of the Audio directory, because all your new songs will share the same Audio directory. You could copy a template CPR to the desktop, this will then prompt you for a new project directory, but that’s not much better than opening Cb first and picking a template.
How about this:
First prep your template, once only -
a) Create a new project based on your chosen template
b) Save it in a proj dir somewhere useful, call it something notable e.g. ‘Template-withPiano’ for example
c) Exit Cubase. You now have a ‘template’ project directory
Then for new projects -
d) Copy and rename the whole template dir (outside Cubase) to when you need a new project
e) Open the new project CPR in the new dir by double clicking the CPR file icon
f) When cubase opens you will be prompted to confirm the ‘change of Audio dir’, accept the new location
g) That’s it, use the project
h) Save, incremental save, save as, etc, all will go to the new proj directory along with the audio
This saves me trawling through directory layers in the Cubase dialogues - I just find it much easier to navigate using Explorer to start with.
I wish Cubase had a special big red button @ startup which automatically recorded anything I played and created tracks for whatever was recorded. That would let instant creativity flow completely freely 
Mike.