Truffle, I will help you. At this point your iPad is pretty much frozen and you do not have many choices to clear it. You seem to have tried all the easy ones (I hope you did them all as you said, and performed them correctly). We may have to wipe your device clean and install an new iOS but this will clear all the setting so you would have to reenter them again and it will take sometime to do so. At least you will end up with a working iPad again. I am assuming that there is nothing wrong with the hardware, this is always a possibility but the way you describe it, I doubt it.
First let's make sure that the easy things are done correctly so I would ask you to indulge me an do them again.
Kill the Mail App completely from the multitasking window by double clicking the Home button fast and then swiping up the Mail preview pane all the way up until it disappears from the screen (you will have to find he Mail pane by scrolling side to side as there are bound to be many preview panes). In fact kill them all, i.e,., swipe all preview panes up until they disappear. The only that should remain is the home page, that one cannot be terminated. Then go to Settings > Mail, Contacts, Calendars > and look under Accounts at the top, locate your Mail account, should be the second item, leave iCloud as-is. Do you have more than one Mail account or just one?. Hopefully just one, click it and at the bottom click Delete Account.
Now do a hard reboot which is totally different than a regular reboot. Hold down the Home and Wake/Sleep buttons at the same time for about 15-20 seconds until the Apple logo appears. Make sure that you start pressing both buttons at the same time and hold them (if you don't all you will do is take a screenshot), Ignore the "Slide to power off" text if it comes up. You will not lose anything at this point.
After the reboot completes go back into Mail, Contacts, Calendars and re-enter your account again by clicking Add Account (again leave iCloud as is). After you enter your credentials NOW try your Mail App again and see if it has cleared the problem. Hope it does at this point because you have not lost anything other than time
If the above does not work you have no choice but to wipe your iPad clean an set it as new. Forget about iTunes for now. First you have to remove the Find my iPad lock as with it on it will not let you reset. Go to Settings > iCloud > Find my iPad > and on the next page turn it off. You will need your Apple ID and PW. After this get out of iCloud, go to General, on the right at the very bottom find Reset > click it and do a ERASE ALL CONTENT AND SETTINGS and follow all the prompts. You need to have your iPad plugged to the charger at this point. After it completes do not restore from either iCloud or iTunes as those backup are now very likely corrupted with the Mail issue. If you restore from them you will restore the problem again and would have wasted your time and mine... When prompted set it up as a NEW iPad and baring hardware issues it will work like when you took it out of the box. The drawback is that you will have to reenter all your settings from scratch but it should now WORK.
I am going to bed now but will check tomorrow to see if you need more help. In any event let me know how it worked....
elcpu