IPad still plays the music when muted – how can I truly turn off sounds

If your iPad is a 3G/Wi-Fi model, then you could conceivably do this through the music app. The reason for the need for 3G is that in order to connect these devices there must be some sort of network present.

That being said, this seems like overkill. If you are using your iPad for music and only music in the car, and if you have the means and desire to pull this off, then I would strongly suggest you simply get a car stereo with a 1/8" input auxiliary jack. This would allow you to plug the iPhone directly into your car stereo, without any other hassle.

Furthermore, streaming data like this is a good way to unnecessarily rack up data usage charges, as you would be transmitting data 100% of the time. Go with the plug-n-play solution: you stream no data at all.

Seriously think about this. Again, it seems a bit like overkill.

iOS does not have an option for this, nor does apple approve apps which alter the status bar. If you want this function you will have to jailbreak your iOS device and install a third party status monitor. As pointed out in another answer, LockInfo does this. SBSettings used to, but no longer does. If you're running an iOS 4.x device, OpenNotifier might be useful as it does this as well as give you pull-down notifications similar to android and iOS 5.

If you're developing for this device, however, you can add this function yourself. As a developer you can run custom code without jailbreaking the device - you just can't submit that code to the app store.

The trick, then, is going to be finding example code for this feature. I expect someone in the jailbreaking community will have some info on what the code needs to do, and if you're lucky you'll find an open source project for it. Then you can compile and sign it yourself, and as long as you keep your ad-hoc license current and recompile/re-sign it yearly, you can use it on your personal devices.

I think a lot of developers overlook the fact that they can run any code they want on their own devices without jailbreaking. They can also distribute such apps in a limited manner to their friends using adhoc distribution as well.

It looks like OpenNotifier is open source, and has the functionality you need:

https://github.com/n00neimp0rtant/OpenNotifier

I don't know much about the status of the project, but it might be usable for you as a developer to bring this functionality to your devices.