Updated : 30/06/2017
NO - You can ask for the marriage to be registered in your home country, but this is not compulsory. As long as you have complied with the law of the country where you got married, the marriage will automatically be recognised throughout the EU – unless it is a same-sex marriage. These are currently only recognised in 13 EU countries: Belgium, Denmark, Finland, France, Iceland, Ireland, Luxembourg, the Netherlands, Norway, Portugal, Spain, Sweden and United Kingdom (excluding Northern Ireland).
NO - It is up to the Italian authorities to decide if you can get married there. Normally they would only let you do this if you had some kind of link to Italy – for instance at least one of you lives there or has Italian citizenship. But if they do, the marriage will be recognised throughout the EU.
Once back in Slovakia, you will have to get any official documents proving your marital status updated, on the basis of your Italian marriage certificate.