Okay. Try the "tried and true" first step suggestion for an iPad that's acting wonky. Do a full reboot: Hold both the power and home buttons until the screen goes dark and the Apple logo then appears (the slide to unlock screen may or may not appear). Once the logo appears, release the buttons. After a time, your homescreen will appear. Please note that it may take some time for the Apple logo to appear or for your homescreen to appear (or both). This is normal. Eventually, the homescreen will appear.
Fixes a large amount of initial iPad troubles...
Yes, an app is really gone if you delete it from the iPad. Except, of course, that it's in iTunes if you've ever done a sync or a backup. Oh, and now you can check purchases in the App Store.
So, yeah you can delete it from the iPad (and from iTunes). But you can't delete the App Store record of it.
Hope that helps.
Marilyn