Spotify Offline Mode: do I need to launch the app within 30 days? (when I'm online) If I use Spotify in Offline Mode on my iPad, I need to go online within 30 days or I will lose the offline songs.

But do I need to open the spotify app when the device does have internet connection, or can I just leave it close, and it will automatically inform the server?

In other words, do I need to take care about it and open the app when I do have connection, or this is done silently in background as soon as I connect to the internet (even with the app close) ?
I think your misinterpreting the intent of that feature. Spotify offline mode basically forces your app to not connect to the Internet, in other words only the songs downloaded to your phone via the 'available offline' switch will play when the feature is enabled. After 30 days the next time you open the app it will disable the feature and connect to the Internet as normal. However, this does nothing to your music, it won't delete your downloaded songs or move it or touch it in any way, it just lets the app start up with internet connectivity allowing you to stream songs. I use this feature too and have no issues with it. 

EDIT straight from their website: