Apex and Missed It
I've tried the latest version of Missed It. It does work, but with a couple of really big flaws. Problem 1: If I clear notifications from the status bar, the count is cleared from Missed It as well. Kind of pointless to have this app if all it does is mirror the notification bar. Problem 2: E-mail Count is wrong. I have more than one account configured in e-mail. I've tried multiple event settings set to all three different settings, but no dice. It appears that the stock e-mail app sends notifications with number of unread mails every time it gets a bunch of new e-mails. But it sends one notification per account. So count as single or multiple probably doesn't work since it sends only one notification if more than one new mails arrive at the same time. Count as value does not work since it sends one notification per account, so the last account notification overwrites previous. Count as value can work if it is additive. Note that Missed It is supposed to provide counts for Nova Prime, but I didn't bother with config since it does not work for me.
The last one that I've tried (and it worked) is Apex Pro. Don't know why I didn't try this first. Works right out of the box. Note that a companion app called Apex Notfiier must be installed. But is all seamless and works great with AOSP dialer, SMS, Calendar and E-Mail. On dialer, it clears the count if I clear the notification bar or open the dialer, which I think is ok. It does the proper thing for SMS, E-mail and Cal and retains the true count as report by the app. It is supposed to work with few other email clients including K9 but not with Whatsapp, facebook etc.
So, I stuck with Apex Pro ($4) in the end. Not sure why this is not native to Android since the badge/unread is very useful.
Hope this helps others.
On a different note, I couldn't find a decent email app that combines exchange 2010 + POP3/IMAP so I had to stick with Stock. Stock works fine, but limited in its user interface and tweakability. Be careful with Emoze as everything is routed through their servers in Israel.