I am using Outlook on my 920 for my company emails. Is it possible to configure it in such way that when I delete an email on my PC Outlook this email will be automatically removed from phone's inbox? Just like Gmail works.
What are the sync settings on your device? If it's set to "when items arrive", it should happen pretty much immediately. Otherwise it will happen exactly when it's supposed to perform a sync.
Notifications in the action center won't, afaik. But the tile clears and should you have set your lock screen to show an icon for that particular address too, it will clear as well.
At my work we use Exchange, with several of us using WP. The WP users report that Outlook on their desktops do not always update when they take action on their phones. Most of us don't use email heavily on our phones since we're primarily at our desks, so I have nothing more scientific than that. We haven't tried to figure out why it happens.
It does both ways. When you delete a mail you delete it at outlook.com. Your Outlook ( phone or Pc) is nothing else then a viewer/editor of Outlook.com. When you delete something on your pc you delete it in outlook.com and you won't see it on your phone. It works the same the other way.