Secenario is the fact that website comes with an admin for website in wordpress and customers are only able to publish/edit their posts.

User panel is diffrent form primary theme (its default grey one).
It is possible to method to write a custom user panel that will use wordpress
login which would seem like primary site, getting only publish/edit/logout functions?

yes, you are able to change the way your dashboard looks.
regarding how to change design for dashboard - have a look here.
regarding how to restrict user use of only song from the dashboard - it can be done easily using the capability manager wordpress plugin.