I've done a wedding at the registrar's office in Edinburgh, so I can offer advice - First, ignore the answer below that mentions the castle as a venue, as you can't be married there unless you're someone special with connections. Not to say you're not special, but, sadly, it won't happen.
The registrar's office is actually very nice and if you want an inexpensive ceremony, I'd recommend it, and save your money for a nice meal afterwards. We had a last minute wedding with a reception at the Balmoral, and the ceremony was lovely. The woman that performed our ceremony did a great job and the room we had it in was very nice. I believe we paid £120 for everything - application, license, ceremony (with 16 guests). The £250 mentioned above is definitely wrong, because I was amazed how reasonable it was for such a lovely venue. It's definitely not a dingy courthouse wedding venue I've seen in the States. It's a small, intimate room that is well-decorated. You have to file your paperwork 2 weeks in advance, so you won't be able to show up and get married right away. At the time you file your paperwork, you can book a ceremony as well. Email or call the Edinburgh registrar's office to see if you can fax in the paperwork. It's a very easy process and they're very helpful, and deal with foreigners getting married here often. You'll need a certificate from Italy declaring you have no reason you can't be married (in some cases you won't need this).
As for witnesses, you MUST have 2 people. I'd recommend asking the office about this when you email them, as they must encounter this problem often. They may offer you 2 of their staff to be your witnesses. If not, why not ask tourists you see outside the registrar's office if they'll do it? It takes 30 minutes. Witnesses shouldn't be hard to find, but they're necessary. I live nearby and I'd be happy to be one of your witnesses!
There is a registrar's office in Central Edinburgh, on the Royal Mile, and I recommend this office. I haven't seen the others, but the cost is the same and it's in the best part of the city in terms of scenery. It's across from the St Gile's Cathedral, which is a beautiful place for after-ceremony photos. If you're getting married in Edinburgh, you will want to do it on the Royal Mile!
It's much more expensive for an approved venue - you have to pay the fees for the application, and pay a registrar to show up, in addition to paying for the venue. You'll also have to pay for decorations, etc, so the registrar's office is definitely the ideal choice.
In terms of property division, you'll be subject to the laws of your own country.
As for after the ceremony, you could hire a car to take you around the city, and then go for a nice meal. If you want something more casual, you could do the open-top bus tour (on a nice day), which is a great way to see the city. If you're willing to spend a bit more on a meal, Number One at the Balmoral is excellent (it's where we went after our wedding), or there's Martin Wishart or the Kitchin in Leith. You can expect to pay around £200 for a 2-person meal (depending on wine). Or, there's The Witchery by the castle, which has a great atmosphere.
If you have any more questions, I can definitely help you, as I have personal experience with registrar's ceremonies in Edinburgh.