Copy of Passport and Lease Contract in Cinque Terre?

Sure, absolutely. Every hotel and B& B in Italy will make a copy of your passport at check in. Emailing a copy is safe.

They are required by law to get the passport info. Think of the 'lease contract' as a letter of understanding especially since neither party can enforce it realistically. They are just making sure you acknowledge the rules.

Yes.

Such requirement stems from art. 109 of the Uniform Code of Public Security Law, an article first introduced in the law by the Mussolini Administration through the Royal Decree no. 773 of 18 June 1931 (and subsequently modified in 1993, 1994, 1995, and 2001). A translation of the main excerpts of the article is provided below:

1. Operators of hotels, and other hosting enterprises, including (...) owners or operators of vacation homes and room renters (...) may host exclusively guests in possession of identity card or other official document proving one’s identity according to current regulations.
2. For foreigners from outside the EU it is sufficient to provide a passport or an equivalent document in compliance with current international agreements, as long as displaying the bearer’s photograph.
3. Within 24 hours from arrival, the aforementioned subjects at paragraph 1 above must communicate to the police headquarters with jurisdiction on the area, the demographic data relative to the hosted guests (....) in accordance with the Ministry of the Interior decrees and the privacy laws.

As you can see Mussolini’s Fascist regime, even then, cared about the safety of tourists.

communications should remain within the airbnb system