You do have system sounds muted. The fact that you have sound in the music app is the indication that the problem is system sounds. You do seem to know how the system sounds can be controlled, but I am posting my explanation just in case. It never hurts to double check.
System sounds can be muted and controlled two different ways. The screen lock rotation can be controlled in the same manner as well.
Settings>General>Use Side Switch to: Mute System sounds. If this option is selected, the switch on the side of the iPad above the volume rocker will mute system sounds.
If you choose Lock Screen Rotation, then the switch locks the screen. If the screen is locked, you will see a lock icon in the upper right corner next to the battery indicator gauge.
If you have the side switch set to lock screen rotation then the system sound control is in the task bar. Double tap the home button and in the task bar at the bottom, swipe all the way to the right. The speaker icon is all the way to the left. Tap on it and system sounds will return.
If you have the side switch set to mute system sounds, then the screen lock rotation can be accessed via the task bar in the same manner as described above.
This support article from Apple explains how the side switch works.
If this still doesn't work, reboot your iPad by holding down on the sleep and home buttons at the same time until the Apple logo appears on the screen. But I do think that system sounds are muted.
Try this - Reset the iPad by holding down on the Sleep and Home buttons at the same time for about 10-15 seconds until the Apple Logo appears - ignore the red slider - let go of the buttons. (This is equivalent to rebooting your computer.)
Check your settings. The iPads have a small switch on the right edge. It can be used as a rotation lock to keep the screen from automatically reorienting itself as you move around, but you need to have the tablet’s settings configured properly. That same switch, right above the volume buttons, can also be set to function instead as a mute button to silence certain types of audio.
If the switch is set to work as a mute button, you can change its purpose to “screen-rotation lock” by tapping the Settings icon on the home screen. On the Settings screen, tap General on the left side, and on the right side of the screen flick down to “Use Side Switch to.” Tap to select Lock Rotation or Mute to set the button’s function. Even if you set the side switch for your preferred use, you can still mute the Mini or lock the screen. Just double-click the Home button, and when the panel of apps appears along the bottom edge of the screen, flick the row from left to right with your finger. Tap the icon on the far left side of the row to either lock the iPad’s screen or mute the iPad’s alerts, notifications and sound effects. Music, podcasts and video are not muted unless you turn the volume all the way down.
iPhone: Can't hear through the receiver or speakers