I tried the app only once - and uninstalled straightaway again because it worked so badly. Maybe has changed by now but I am not going to try again.
By the time we see some new listings, they are over 12 hours old.
You say you do not have the app installed. Then of course you do not get any push notifications.
In the web interface, favouriting does not seem to have any affect. Or it might be my browser settings (I switch off pop-ups), but I have never noticed anything.
You only get push notifications through the app. So you will need to install and use it to see the not-really instant notification for favourited listings and the much more delayed search updates.