If you are citizens of EU countries, EEA countries such as Iceland, Liechtenstein, and Norway, and stateless persons or refugees who have resided in Italy for more than 3 years there are no special requirements if you are seeking to buy property in Italy.
If you are a Foreigner residing in Italy legally and your family members are in good standing or are a stateless person/s who has lived in Italy for longer than 3 years then you are eligible to buy property in Italy without the need for verification of a condition of reciprocity if you have a residence permit for:

Employment or self-employment
Sole proprietorship
Family reasons
Humanitarian reasons
Study
And if you have a long-term EC residence card.

Foreigners who do not reside in Italy can buy property as long as there is an international treaty that permits a material condition of reciprocity between your country of origin and Italy. Reciprocity means that Italians are allowed to buy a house in the foreigner’s country of origin. The Ministry of Foreign Affairs periodically verifies reciprocity between Italy and other countries such as the United States or the UK. Please note that notaries are not allowed to draft any forms of property buying contracts involving citizens from “non-reciprocity” countries.

Yes. some towns/villages are still selling homes for €1 but there are a few things to consider before buying one:
These properties are normally in remote areas
They will usually need to be renovated considerably and this must be completed by a certain date
Resale value may not increase much despite the investment in bringing the property up to standard
Other aspects that one should consider before buying one of these properties include:
Due to their location employment opportunities may be non-existent
Internet connection could be an issue
Medical facilities may not be in easy reach of the property’s location
Purchasing one of these properties as a Non-EU citizen does not give you any special immigration status
Before buying one of these properties, it is highly advisable to go on a viewing trip and to engage the services of local experts such as a lawyer, accountant, and surveyor to avoid any unpleasant surprises down the line.
For more information on these programs, you should contact the Town Hall directly in the towns that are offering these homes for sale. Estate agents do not offer these properties.

That depends on what you term cheap. The type, location and condition of a property determine the price. Generally real estate in cities is more expensive than in the countryside.
The average cost for property in Italy is €1,719 per square meter (sq. m.) which is average compared to other European nations.

You should not be looking at the cheapest area but focus first of all on what type of property and what you need to be near to and the type of amenities/facilities you want to be nearby. Doing that will allow you to find a property that will suit your needs. Focussing on the cheapest area will just waste your time and may dishearten you in your search.

You can as long as you have the appropriate visa, details of where you are living ie rental agreement or contract of property purchase and proof of funds that allow you a comfortable standard of living in Italy.
Details can be found at
The Italian Ministry of Foreign Affairs and International Cooperation and the Italian Consulate (https://vistoperitalia.esteri.it/home/en) have more information about:

Visas, including intra-corporate transfers
Work and residence permits
Supporting documentation
Other conditions

Once again, this depends on the type of visa you have and whether a Schengen national. Non-Schengen national can stay for up to 90 days as a “tourist”. For longer stays, you need to look at Student visas, Work visas, Self-employed work visas, or Elective Resident visas.
Further details at Details can be found at
The Italian Ministry of Foreign Affairs and International Cooperation and the Italian Consulate https://vistoperitalia.esteri.it/home/en

Simple answer is yes but there are procedures which need to be followed and put in place.
Americans staying in Italy for more than three (3) months are considered residents. This includes Americans who will work or transact business and persons who want to simply live in Italy.

Americans must have an entrance visa which should be obtained at an Italian consulate before coming to Italy, in order to remain in Italy for more than three months and gain resident status. This procedure will take several weeks to complete so it is advisable to apply well in advance of the departure date. The visa will be granted only for the time indicated on the application. Americans already in Italy without a visa will have to leave Italy to obtain one before they will be able to gain resident status.

Upon arrival in Italy, the American must obtain a permesso di soggiorno (permit of stay). As of December 11, 2006, an application “kit” for the permesso di soggiorno may be requested from one of 14,000 national post offices (Poste Italiane). The kit must then be returned to one of 5,332 designated Post Office acceptance locations. It is important that applicants keep a copy of the receipt issued by the Post Office. Additional information may be obtained from an Italian immigration website via the Internet at: http://www.portaleimmigrazione.it. Within 20 days of receiving the permit to stay in Italy, Americans must go to the local Vital Statistics Bureau, Anagrafe of the Comune, to apply for residency. It generally takes one to two months to receive the certificate of residence (Certificato di Residenza).

In addition, as of March 10, 2012, under Italian law (Presidential Decree no. 179 of September 14, 2011) all American citizens who request an Italian permit of stay for more than 12 months are required to sign an “integration agreement” (accordo di integrazione) at the local prefecture (sportello unico per l’immigrazione) or at the provincial police headquarters (questura). By signing this agreement, Americans are compelled to achieve specific integration goals (e.g. acquire an adequate knowledge of the Italian language, of the Italian civil structure and culture) and accumulate a total of thirty credits. Simply signing the agreement will secure the first sixteen points. The remaining fourteen points must be earned over the next two years by taking classes or passing a test in the Italian language, on the civil structure and culture, and by successfully completing a variety of other requirements. Failure to complete the integration agreement with the required credits is punishable by expulsion from Italy. Additional information may be obtained from the Italian Ministry of Interior’s website at http://www.interno.gov.it/it/temi/immigrazione-e-asilo/modalita-dingresso/accordo-integrazione-straniero-richiede-permesso-soggiorno.

Since time immemorial visitors to Italy have fallen in love with the land, the architecture, the culture, the cuisine. Some cherish these memories, carrying them in their heart for the rest of their lives; no matter where they are on the globe, their spirit slumbers in Italy. Others make Italy their own—their home away from home—by securing their own apartment, villa, house, or estate in the birthplace of Western civilization, the country of love.