I do have Android Pay enabled so I already know about that , but it was happening before I enabled Android Pay. I was not aware of the problem Exchange accounts could create. I have my college email setup through the stock email app which is an exchange account. I have another email client (Type Mail) with my college email, as well as my GMail and Yahoo setup in it. So if I disable the stock app it won't affect my other email client with email synching will it?
It depends. Not all Exchange accounts enforce this (mostly just corporate accounts). If that would be the case, the Exchange account would appear as a Device Administrator. In theory, if the Exchange account enforces this, no matter what app you use, it would ask to add itself as an administrator or it wouldn't let you configure/use it.
As for the account log out... not sure. It shouldn't as the Account section is separate from the general phone settings, so I believe it would only (worst case scenario) turn everything back to default. BUT if you selected to do so and Settings are backed up to your Google Account, it should, in theory, just re-sync and re-apply whatever you had, minus the bugs.
You can always try to clear cache first (if available).
I remember someone saying that they changed their lockscreen method and it suddenly made NONE available (like going from PIN to Pattern).