The only time this happens to me is when my ISP has problems with their email system. Have you checked on the BT Service Status site to see if it is a problem at their end.
My btinternet mail went through a phase of asking me to log in everytime I clicked the shortcut. Then it stopped, for some reason. I did read that once every 6 months you have to sign in via the BTYahoo website to keep the email address active. Maybe it`s something to do with that.
I always have to log into my various email accounts, and I think that's right. I think it's dangerous to remain logged in, just as I wouldn't want to remain logged into internet banking account. I also recommend changing the password from time to time
I only have one e-mail address which worked trouble free until I got a message saying the user name and password were not recognised despite no cahnges being made on my part. I did go into the BT site and changed passwords making doubly sure that my computer settings were correct.