The Hyundai dealer doesn't know what to do because they are very afraid when there are ghosts in a customers' car and they want no part of a car radio exorcism.
That said, it's quite difficult to understand exactly what you mean is happening in your car and car radio. Do I understand correctly that you leave your car radio on when you shut the car off, but when you start the car again, the radio is not playing, and it should be? But later it just starts playing again all by itself? And this only happens sometimes, not every time? And when it does that, there are beeping sounds? How many times does it beep? Is it the same number of times every time? What about the beeping "doesn't make the music sound good"? In addition, when you turn the radio volume up or down, it goes up or down, but it takes a while?
I have never heard of there being a delay between turning on a radio and the radio coming on, or of a delay when turning the volume up or down and it actually going up or down. It doesn't make sense. The radio, cd player, XM etc. are electronic. They don't "choose" when to react. I am unaware of anything at all that could make them react moments later than when the input is given directing them to go on, get louder or get quieter. This is where the ghost theory comes in. ;)
I don't mean to be impolite, but that's a whole lot of crazy stuff going on, and it's inconsistent? AND across all of your car sound options? When that many things go wrong all at once, it's not good. I wonder if it even CAN be fixed. It seems like it's not just one problem, but a bunch of problems, each of which would have to be found, figured out, then fixed if possible. In my estimation, that would cost approximately what a new 2011 Hyundai Tucson Limited would cost. Perhaps the entire system needs to be rewired?
The dealer can't help because this is very complicated and really really strange, Guru V7Q2G. Don't be mad at them.
Your car sound systems may be too messed up to save. Think about replacing them. Scour the internet on Google to see if anyone else has ever had this happen with your brand name sound systems.