The iTunes symbol with the cord means you'll have to put your device into recovery mode. Here are the instructions:
If you can't update or restore your iPhone, iPad, or iPod touch - Apple Support
Your description with the red battery sounds like it's low on power. Let it charge for at least 30 minutes, before you use recovery mode.
Usually, it's possible to turn off the iPad by using the Power button only. If you press Home and Power button, you'll reset/reboot your iPad, which will turn it on automatically after the reset, but it won't shut down your device.
Another method to power off the iPad without any buttons, would be to use Assistive Touch. You can find it in Settings - Accessibility. When you enable the feature, a semi-transparent button will appear.
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Tapping it once, will show you six choices. One of these should be "Device". Tap it, and then you should find "Lock Screen". When you tap and hold this for some time, you'll get the slider with "slide to power off", as usual when you turn off your device.