After I log in to the OJS, when I click on any of the following links
Site Settings
Hosted Journals
Languages
Authentication Sources
Categories
System Information
Expire User Sessions
Clear Data Caches
Clear Template Cache
Clear Scheduled Task Execution Logs
Merge Users
I’m forced to log out. Then I have to enter the username and password twice (Yes, twice), I can go on with the clicking.
This sounds like you might be changing the domain when following internal links. It could be as unobtrusive as myjournalsite.tld → www.myjournalsite.tld.
Compare the link URLs with the URL in your browser’s bar.
Once difference which IE exhibits from other browsers is that cookies are shared across subdomains (see Q3). Since you describe access limited strictly to the www subdomain, it seems like this shouldn’t be relevant, unless we’re missing something.
You might try clearing cookies for IE to see if that makes a difference; also, try IE on a different computer to see if it really is always IE, or just your IE.