There is testing work still going on on Stack Exchange in general, and that semms to have some side effects. For some time, all SE sites were in read only mode, that is no longer the case. But looking at the status page, there is no message stating the testing work is done. But it does state the work is in the process of finishing.
For the status ...
The instances we've seen this in were browser crashes or a tab left open with the inbox expanded - this reads notifications as soon as they are received. Nothing has changed on our side since the move when real-time functionality was re-enabled (about an hour after the move).
To expand what I mentioned in the comments, you get notifications of unread inbox messages, not all inbox messages.
(Un)fortunately, clicking on the MultiCollider (aka the Stack Exchange notification dropdown menu) marks all messages as read, which is what I assume is causing you to not get any emails.
You don't get emails by default, unless you've specifically enabled that when you ask a question.
For comments & other notifications, you can click on email settings and enable the email notifications for your unread notifications.
Oh well... this seems to be stupid(and it is). It is related with the new questions that keep appearing when the page is not refreshed.
"One question with new activity"
The badge notification, only was loaded when the page was refreshed, and that explains why did it not appear (That minute was spent reading question titles before refreshing).