Thanks for the update and I'm glad it's solved. The icon in the upper right (circled) should not have had any bearing on the problem though, since it only appears when hovering your cursor over it. Visual feedback elements that appear and disappear in that manner is not the most well thought out in terms of user interface in my opinion, but that's besides the point. I wonder if it is related to this excerpt from If you can’t send or receive email on your Mac - Apple Support:
"You might see a status icon such as a lightning bolt or warning symbol next to your account's inbox in the Mail sidebar, or in the upper-right corner of the Mail window."
Clicking those icons either causes an informative message to appear, or it brings the affected accounts back on line. That would make sense.