Whether I am trying to sign into this account using the Windows 10 Mail app or through my institution's website via Edge, I cannot login into my account. In addition I get a very uninformative error message which just show a "#" on the tip right corner of the screen.

I've confirmed that I can login to my institution's Office365 email using internet explorer but I'd much rather use the mail app or Edge. I've seen that others are having the same issue for their university email accounts, but the only solution provided was to set IE as the default browser which in my opinion is not a solution. Are there any other options to try?

I recommend that you post this in the Office 365 Exchange Online Forum:  http://community.office365.com/en-us/forums/158.aspx

Hi,

We can do the following points to narrow down the issue to Windows 10 side or Exchange Online side:

1. Confirm whether Windows 8 or 8.1 users can use their Exchange account in Mail app or not.

2. Change a new Windows 10 computer to have a try.

If the Windows 8.1 users can use Mail app with your Office 365 account properly, the issue should not be related to Exchange server. If your account works in a new Windows 10, please disable any third-party program installed in your original Windows 10 or perform a clean boot to check whether the issue persists. Additionally, we can ask a question in Windows 10 forum for more help:

https://social.technet.microsoft.com/Forums/en-US/home?category=Windows10ITPro      

If the issue continues in all Windows clients, please ask for help in Exchange Online forum as ED suggested.

Regards,