Your experience is not typical. There are no widespread reports of problems with 6.1.3.
It seems likely that you could demonstrate the problems any time you wanted. So please take your iPad into a AppleStore (book yourself a Genius appointment) or phone Apple and report your problems to them. They may be able to help, but it would certainly be helpful for a support technician to take a look.
Use Apple Configurator to downgrade. Its supported. It will at least allow you to confirm whether indeed iOS 6.1.3 is the culpret.
I definitly agree with Simon, take it to an Apple Store, they want you to be happy with their product and are willing to do a lot for you!
Message was edited by: AndyBeaver