Is the deck app capable of sending email/push notifications?

I am trying to get email notifications for Deck to work but I am not sure if this is a feature or not. I have set up mail server and am able to receive sharing and test emails. Under the user settings I have ticked both for
Changes in the Deck app
A comment was created on a card
A card description inside the Deck app has been changed

Should I get notifications when a card is assigned to me commented on etc?

Yes. You need to turn on notifications in your account settings first, though.