I am a Ukrainian citizen who is studying in Italy. Several weeks ago I have applied for renewal of my permesso di soggiorno, so at the moment I have expired permesso di soggiorno and new ricevuta di soggiorno. I also have a biometric passport which allows to travel within Schengen zone for 90 days (since 2017 there is a visa-free regime between Ukraine and the countries of Schengen Zone)). Therefore I'd like to know am I able to travel within Schengen zone with ricevuta and biometric passport (I suppose yes, but I haven't found any official document where it is written extensively. Also, I know that if you're holding a temporary residence permit issued by Germany/Slovakia you can travel within Schengen zone with a biometric passport). I'd also like to know am I able to leave and re-enter to Schengen zone through other Schengen countries with those documents.
No, the receipt is not a valid travel document. You must wait for a new plastic anti-counterfeit residence permit card to travel to other countries, or use the rest of the 90 Schengen days. In the latter case, make sure you have an entry stamp on your passport.
The receipt can be used only to enter Italy from Ukraine, not to travel to other Schengen countries.
I said about the entry stamp because you won't get a stamp by entering Italy with the residence permit, but the gate agent at the airport will thoroughly check your passport for stamps.