Recursive Apple ID password procedure

I think this may be a phishing. I advise you to check the 3 following key points:

1. Within the headers of the 2 E-mails you received, verify they are really originating from apple.com .
2. Verify the included URL is really within the apple.com domain.
3. Within the headers, verify there isn't any Return-Path: outside of apple.com .

Before you do anything, make a backup of the iPad to a computer if you can.

Also, convincing a school or company to reinstate an email for long enough to recover an account is often doable with patience and some cookies or a pie to the person that’s inconvenienced or needs to approve such a request.

That being said, In my experience @books453 is right and iOS asks what you want to do with photos and other iCloud data when you sign out: so you should be able to remove them from the device or leave them there as long as you aren’t wiping the device.

Since they remain on a device you control, you are able to add to a new iCloud account when you sign in with it. However, I don't know whether this applies to data stored in apps and games, such as game scores.

When you say I changed his email address for iCloud via my laptop when I set up his phone, it sounds like you actually created a new iCloud account for him. If you had in fact changed the email address associated with his iCloud account then I don't think Apple would still be sending confirmation emails to the old email address, as you say many other people have already told you (sorry to repeat it!). To verify whether the new email address is attached to his old iCloud account, log in to http://icloud.com/ with the new email address and see whether the photos and other data from the iPad are are available there - if they are, I would be confident that the replacement of the email address has worked and therefore it should be safe to sign out of the old address and sign in to the new address.

If this was my device and data, I would make an appointment with the Genius Bar at a time when the Apple store won't be busy and ask them to walk through the process with me. Your son's new iPhone should entitle him to at least one session of personal setup assistance and technical support. And don't accept what the first Genius tells you if it's not what you want - get a second opinion, as this is a tricky case that not all staff may be familiar with.

EDIT:

Sorry, I missed that in your question you mentioned you can't even sign out of Find my iPad, which is preventing you from signing out of iCloud even if you want to. I assume that when you tried to go through the process of signing out of iCloud, you eventually hit this prompt:

And based on the confirmation email I received after turning off Find my iPad, you may not even be able to restore or reactivate your iPad unless you can turn off Find my iPad first, which you can't because Apple isn't accepting the password for the old email address:

Presumably this is part of Apple's anti-theft measures that try to make stolen iPhones and iPads useless.

Is there any chance you can visit an Apple Store? If not, I have found their on-line chat people to be very helpful. Does your iPad show up as a registered device under your son's new iCloud email address at https://supportprofile.apple.com/ ? If so, I would hope that Apple's support team would at least try to help you.