SO many problems since the iOS 9.3 iPad update. Grrrrrr!

It is a royal mess unfortunately and you are one of many having serious issues. Take a look under Settings > General > About > Version and please tell us what version downloaded. We may be able to offer some suggestions depending on what you have.

Also please take a look at this summary by a very experienced user.

For those having problems with the iOS 9.3 update please read...

That is the latest version. Please take a look at this article. It may help.

http://osxdaily.com/2016/03/25/troubleshooting-ios-9-3-update-problems/

Most users are reporting serious issues with hyperlinks in Safari and Mail and are awaiting a further fix beyond 237. But your iPad appears to be in worse condition. You could try Recovery Mode which will erase your device but I cannot tell you with confidence that it will resolve. May want to first read the link above and the post by Demo (my previous link).

There is also DFU mode that should be tried as a last resort.

You may have no choice but to take your iPad to an Apple Store. I would not delete your iPad from iCloud.com as I don't think that would help.

You could try DFU mode.

http://www.imore.com/how-to-iphone-ipad-dfu-mode

Demo, the highly experienced user I mentioned above, is likely to see this later today and may offer better advice.

I understand your frustration.... hang in there, I am trying to get in touch with Demo.

Don't throw the iPad out the window. It's frustrating, but it's not worth ruining the device.

Can you get into the settings app? If you can, reset all settings and then force restart again. Settings>General>Reset>Reset all settings. You will have to enter all of your settings again but it's not nearly as much work as restorping the iPad.

Recovery mode should negate the need to enter a password anywhere. It does erase the device so hopefully you have backup. Is this how you are trying to restore?

If you can't update or restore your iPhone, iPad, or iPod touch - Apple Support

Put your device in recovery mode and set it up again

1. Connect your device to your computer and open iTunes. If you don't have a computer, borrow one from a friend or go to an Apple Retail Store or Apple Authorized Service Provider for help.
2. While your device is connected, force restart it: Press and hold both the Sleep/Wake and Home button for at least 10 seconds, and don't release when you see the Apple logo. Keep holding until you see the recovery mode screen.
3. When you see the option to Restore or Update, choose Update. iTunes will try to reinstall iOS without erasing your data. Wait while iTunes downloads the software for your device.
   If the download takes more than 15 minutes and your device exits recovery mode, you'll need to repeat these steps and choose Restore instead of Update when you get back to this step.
4. After the Update or Restore completes, set up your device. If you previously backed up your device with iTunes or iCloud and had to restore your device in step 3, you might be able to restore from your backup during setup.

OK. I will be here checking in.

It might not work the first time, give it a few tries.

Good luck.