I recommend creating a new user account (System Preferences > Users & Groups), logging into that user account, and conducting trials from that account.
In particular, set up just one email account from the new user account. Confirm that the email account works. Then proceed to the next email account. Confirm that it works. If something doesn't work, stop at that point and tell us all the steps that lead to that point.
I also recommend using the
Mail Connection Doctor window rather than having Mail.app check for mail. One reason is that you can have it show logs, which should provide more details, and you can also copy and paste into a post here.
The way you're doing it now, it's difficult to tell if the problem lies with the Mail app, Mail's preferences, the email account info, or the user account, or maybe something else. By starting from a virgin user account, and adding only one email account at a time, you break the problem down into single observable pieces. This makes it easier to work out what's happening, rather than trying to observe multiple pieces, any one of which may be malfunctioning, or which may be malfunctioning interdependently.