AT this point, the only other thing that I would suggest is to erase and start over meaning backup, restore to factory settings using iTunes, and restore the backup. If that fails, restore as new and rebuild the device. Its not as intimidating as it seems. I have done it several times with several devices. I just went through it with my iPad 3, after I downloaded iOS 8. The performance was lacking so I did a clean install, but also restored my backup. It helped, but the updates to iOS 8.1.1 helped even more.
Using third party mail apps isn't a bad idea at all. I use the Xfinity Connect app for my Comcast email. It has many more features than the built in mail app, so it's great for my Comcast account. However, I also have two AOL accounts (one is a work account that goes, way, way back) and the other is a MS Exchange account through my Comcast Business account, so I still use the mail app to keep all accounts in one place.
What bothers me about resorting to a third party app is that is just a workaround and I would really like to see you resolve the problem because it could lead to something even worse down the road. Having said that, if that's what you have to do, that's what you have to do. you could just continue to access your email on the server using Safari or another browser, but if downloading another email app makes it easier for you, I would probably choose that route myself.
There must be something corrupt somewhere in a setting on the iPad that's causing this and I was hoping that resetting all settings would fix it. You did try closing all apps and restarting the iPad at some point didn't you?