My friend is deported and banned from entering Norway for years. But he has a Carta Soggiorno here in Italy. Can he still travel in other EU countries? How can he check his status with other EU countries before visiting them? He is a non-EU citizen.
It is up to UDI (Norwegian Immigration) to determine if they would be registered in the Schengen Information System (SIS) database:
"The UDI will also decide whether you will be subject to a prohibition against entry to all the Schengen countries. In that case, you will be registered in the database Schengen Information System (SIS)."
If information is entered into SIS by UID, it is up to the individual country to determine their appropriate course of action (this is why he can have residence in Italy, but possibly be barred from other Schengen countries). Your friend does, however, have the right to request that information, if any, is entered in SIS by UID.
"If you wish to find out what information is registered about you, or ask for incorrect information to be corrected or deleted, you must contact the National Criminal Investigation Service (NCIS, or Kripos in Norwegian). You will find more information about how to proceed and the request form which you need to fill in at the NCIS website (external website)."