I was also having the same problem with the standard email app, i.e. not getting notifications. I had my work Microsoft exchange email, Hotmail and yahoo all on the standard email app. I then deleted Hotmail & yahoo from the standard app and set up a third party app to run them.
But after reading threads that Gmail could run Hotmail and yahoo accounts, I decided to delete this 3rd party app (Microsoft outlook), and start to set up my Gmail app to run Hotmail and Yahoo. Before I added Hotmail & yahoo, I was configuring Gmail (as I realized that I do not get notifications for this email too) and it asked my to sync all data and emails, so I selected yes, then all of a sudden I started getting notifications for my work exchange email on the standard email app.
So I then went back to this, added my Hotmail & Yahoo accounts, sent some test emails and I started getting these pushed through and notifications coming up at the top and locked screens.
SUMMARY
Some how it seems the Gmail default app was preventing/ overriding the Samsung app's notification settings. Hope this helps people.