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Usually recognized refugees as well as asylum seekers have the right to get married. This website offers an overview of the regulations in Italy:
I’ll give you a short summary based on my research:
Your fiancé needs to provide a valid passport and an authorisation or marriage eligibility certificate for foreign national citizens. The most important document he (most likely) needs to hand in is the so-called “nulla osta”. It’s a document that states there are no obstacles to marriage. It is issued by the competent authority of the country of origin (consulate or embassy).
You (being from the US) have to provide a sworn declaration made at an American consulate with signatures legalised at the prefecture's office, an attested affidavit expressed in front of a unified court and a birth certificate (instead of a “nulla osta”).
Unfortunately the information that I found through the website linked above is not very detailed. To find out about the exact regulations and procedures you can get in contact with the local registry office or civil status office. They probably know best.
Concerning your second question: As far as I can see you will not get a residence permit for Italy through the marriage directly. Usually you only get it if you marry a citizen of the country. Unfortunately your fiancé being a refugee is not on a par with Italian citizens in certain legal aspects like that.
But you can try to obtain it through family reunion. First of all your partner needs to apply for an authorisations for family reunification at Immigration Office (Sportello Unico per l'Immigrazione). He can do that online. This link will guide you there: http://www.pratomigranti.it/en/documenti/ricongiungimento/richiesta-online/pagina60.html
Here you will find more information about the authorisation: http://www.pratomigranti.it/en/documenti/ricongiungimento/nulla-osta/pagina58.html
If that’s successful you will receive an entrance visa. After getting to Italy you can then apply for a residence permit for family reasons.
In some aspects refugees and asylum seekers are privileged when it comes to family reunification: for instance they don’t need to prove ownership of requirements such as suitable housing and income. That will make it easier for you.
You can find extensive information about this topic here: