I have the feeling this is one of those things that would be easily diagnosed in person but might be tough online. You don't have Do Not Disturb enabled, do you? (It's in Settings.) If you have an Apple Store nearby it would probably be easiest to just drop in and ask about it.
What happens when you start a song? That detail about not being able to turn up the sound makes it sound like it could be an issue with the headphone jack--like the phone thinks you have headphones plugged in...try putting some in, playing music, stopping it, and taking the headphones out.