If you have a backup in icloud - delete it using Settings\icloud\Storage and backup. once done turn off Backup to icloud.
Open iTunes - connect by cable - allow any sync to finish. Under the main menu navigate to Edit\Preferences click on the Devices tab. Check that <Prevent ipods, iPhones and ipads, syncing automatically> is ticked. Click any device backups appearring and delete them. Then click on Reset Sync history at the bottom of this page.
On your iPad navigate to Settings\General\Reset - first click on Reset Home page, then after it finishes click on reset All settings.
Close the Advanced tab and dialogues in iTunes.
On your iPad navigate to Settings\General\restrictions - disable restrictions.
In Settings\Notifications remove any non factory default Apps from the notification centre.
Next navigate to Settings\Usage\Storage and tap on Show All Apps- tell me what the total storage is given at the top of the page (and also your device size eg 16\64 Gb). Have a look through the list and see if there are any with over say 500mb of data.
The reason I ask is because with the backups deleted I would suggest unsyncing and deleting all backup data associated with the Apps - unless there are some games for instance whose progress you really do not want to lose.
I think you have a memory limit input/output lock that is now blocking further App activity.
Unless we can pinpoint the App causing the trouble - checking crash logs may help pinpoint the offending app though with all apps crashing it may be a case rof working backwards to find the one that first caused the error and working from there.
Come back with the info I have asked for - then we can have a look at the logs - in the meantime dont initiate any syncing or backups or try to download or run anything other than your factory Apps.
Regards, Ct