My iPad mail inbox keeps spontaneously deleting messages or something to make them disappear. They are not in trash..just gone.

Try quitting Mail like this

Go to the Home screen and double click the Home button. That will reveal the row of recently used apps at the bottom of the screen. Tap and hold on the app in question until it wiggles and displays a minus sign. Tap the minus sign to actually quit the app. Then tap anywhere on the screen above that bottom row to return the screen to normal. Then restart the app and see if it works normally.

Then reboot your device. Press and hold the Home and Sleep buttons simultaneously ignoring the red slider until the Apple logo appears. Let go of the buttons and let the device restart. See if that fixes your problem.