He is young, probably has little or no office work experience so as the other answer suggests it is up to you and your colleagues to show him what is OK and what isn't in the work place, that is after all what he is there for. To get experience.
If his boss is too nice to say something, take ownership of the situation and make yourself the one who does. Ask him to turn it down as it is affecting your concentration which provides a little justification as to why it isn't ok. If you can hear it too then it's obviously too loud (defeating the purpose of headphones!).
Distracted employees are not productive ones and can affect project work and overall productivity, this I'm sure will fly with management if you point this out to him.
The Intern also acts like he owns the place, He comes at 11 and
already leaves at 3-4 sometimes. We have until now never gotten a
finished project from him.
We actually had this in my team right now and my boss openly mocks the person in question for being lazy and not pulling his weight for a while to see how he reacted and he started to correct it. Slowly slipping back he then took him aside and had a direct conversation about it, which sorted this out. Maybe this could be something that helps.
You are not being 'that guy', you are protecting yourself, your work and your productivity. If you are constantly distracted and it affects your work it will be you that has to answer for it if mistakes are made etc and simply blaming the interns music probably won't cut it.
However if you've been vocal about it and people have seen you ask him then others may even follow suit or its seen that you've actually tried to do something about your distraction while also mentoring the intern about the work place dos and donts.
Also he recently started singing aggressively and moaning while having
these ridiculously big Headphones on. Any of you have some advice on
how to handle this?
In the same way, make him aware of it and ask him to stop.
If he gets confrontational about it, personally id explain that it is causing a distraction to project work which isn't good for the business and hint you will need to have words with his manager if it continues. Especially when you are helping him get a leg up on a career and he will potentially ask for a reference.