I'm developing React-native app (iOS & Android).
Server connected through web-socket. Server sends notifications about some specific events. App suppose to notify user (using alerts). What if I need to notify user even if app is running in background or closed - how can i do that? Push notifications with this library will help in cases when app is in background, right? But it will not work if app is closed? So, how can i notify user if app is in background or closed?
The library you referenced will indeed allow you to notify users even if the app is closed as it utilised Push Notifications. These are delivered to your users' devices no matter the state of your app.
When a user opens a push notification they are taken to your app. This Push Notifications tutorial is a good introduction to the system.