I am downloading apps on my iPad off of the App Store and they are downloading on the App Store but aren't actually coming out on my device.
It says they have downloaded on the App Store but I have looked everywhere and it's not there.
Something is wonky with your iPad.
First try rebooting the iPad. Hold down the power button until you see the spinning cog wheel and your iPad shuts down. Then hold the power button again until it reboots.
If that doesn't work, then try syncing your iPad to iTunes. If the apps don't appear in iTunes, and on your iPad, then use iTunes to download them, and then sync them over.
If that doesn't work, then place your iPad in restore mode by holding down the home button while then plugging into cable attached to your computer and restore your iPad with iTunes. If after iTunes reinstalls the system and syncs your apps, these apps do not appear, then...
try restoring your iPad without restoring the last backup. If after restoring and syncing, these apps don't appear, then...
seek Apple Genius at Genius Bar at nearest Apple Store.