With time, I believe I have better understood the problem. I cannot say for sure, but I believe that the Mail app will actually notify me instantly when I receive mail, BUT will only do so when messages arrive in my inbox. When messages arrive in another folder, via a rule set on Outlook.com, I am never notified, and those new messages won't appear on the live tile either.
For example, right now, I have at least 8 new messages in various folders, but I have only received a single notification, and the live tile only tells me I have one new message: the one in my inbox.
Before, I used Windows Live Messenger to receive instant new email notifications. (And in its last months it was pretty much the only thing I used it for). Back then, I noticed that Mail (the Modern UI app) was always late by quite a few minutes, while WLM gave me instant notifications.
Now that Messenger is dead, it's a bit of a problem. I usually turn on email notifications for private messages on the forums I frequent, so that I instantly know when I receive one if I'm in front of the computer. But it's not working very well now: Mail's notifications are quite late, and sometimes it just decides not to give me any.
I think this might be related to how my email account is set up (some mail automatically goes in one or another folder instead of the inbox), but I'm not sure. Messenger didn't care about that, after all.
So I would like to ask, why is Mail so late? Why does it skip some notifications?
Is there a way to have Mail give me instant new email notifications?
If not, what would be a simple way to get these notifications instantly, using something else? Please note that I use Outlook.com, not a local mail client, so don't look for ways to configure Office Outlook or anything like that. If possible, if I could have to NOT install another local mail client (because the Modern UI app is one) and thus end up having my mail on my computer in two copies, it would be great.
Lastly, there is a secondary problem. In Windows 8, all toast notifications from Modern UI apps make the same sound. Also, they only stay on the screen for a relatively short time, and aren't stored anywhere (there is a "notifications" thing in the Settings bar, but I have no idea what it's for, since it's always empty).
So if a notification pops up while I'm not looking at the screen and I'm too slow in coming to see it, I have no idea whether it's a Calendar event reminder or new mail. Is there a way to customize the sound from Mail so it's different from Calendar notifications ?