What happened to the Apple Remote that came with your Mac Mini? Or, if you don't have one, you can buy them for $20; you can also control the Mini with any Universal remote control, via any Bluetooth device including iOS devices and Bluetooth Keyboards and Mice, or via iOS and WiFi.
A Screen Share is not exactly the same thing as controlling the audio via an audio application like Audiviranna, nor is it the same as controlling the audio on a Mac via an iOS device like an iPhone or iPad.
Screen Sharing is built in to MacOS. Do you have a monitor attached to the Mini? What do you mean by "iTunes Screen Share program"?
Sorry, I'm not sure what exactly you want to do. Also it seems you are using language that means one thing to MacOS/iOS and another thing to you. Can you be more specific, and then maybe we can help.