I've had the Galaxy S3 Mini for about two weeks and I can't take screenshots anymore. I hold down the home and power button but it just takes me to the task manager window. I also can't seem to be able to turn off or reset my phone.
It's been a while since I stopped using S3 Mini but wasn't long-pressing the home button calling the task manager anyway? If you what you mean by saying that "you can't reboot or turn off your phone" is that power button is unresponsive, then it means it is broken. You can root your phone and use apps to change the button configuration for screenshotting.
Have you gone into Settings and turned Motion Settings on? In addition, you have to be precise. Press and hold the home button and power button at the same exact time and hold it for a couple of seconds. About two seconds when holding but press them at the same time.
Also, depending on your Android version: Volume Down and Power Button simultaneously should work [press and hold]