First, don't get discouraged! Good for you for starting the search.
There are plenty of dogs in need of a good home; you might just have to look a little farther afield.
Each rescue organization makes it's own rules, with a eye to what they feel is best for the dogs in their care. Some are very strict, some far too lax (IMHO). If you speak to a few different Tierheim you will get a feel for differing procedures.
Here is one of my experiences:
I fell in love with an elderly collie who could barely walk... But, because my house has stairs up to the garden, the Tierheim was hesitant to allow me to adopt him - they felt he could not manage. Fair enough. But... I just
knew that this dog was meant to be part of my family. So, I asked to meet the shelter manager. I gave her a detailed write-up of my experience, gave her letters from my vets attesting to how I have cared for my other dogs, wrote up my plans for how I would manage this dog's day to day needs, showed her plans to build a ramp if necessary, brought along the photo album of my other happy handicapped mutts... in short, I pulled out all the stops to convince the shelter that I could provide exactly the home this poor old boy needed.
He came home with me the next weekend.

(Although, I often wonder if their change of heart was due to the fact that he suddenly needed surgery, rather than to my eloquence...

)
I have also adopted a dog from Italy. There are, sadly, thousands of dogs languishing in shelters in Italy - most of whom never make it out.

Get in touch with one of the Swiss organizations working in Italy, or go to any of the Italian rescues on your own. To bring a dog back from Italy you will need an EU passport (rabies vaccs, etc.) For import regs, see the BVET site:
http://www.bvet.admin.ch/00386/index.html?lang=en
Two Swiss groups working in Italy:
http://www.sos-strassenhunde.ch/neu/index.php http://www.tierhilfe-italien.ch/Home.htm
And, Oasis des Veterans also works in Italy - they are dedicated to rehoming older dogs:
http://www.oasis-des-veterans.com/index2.html
You could also consider rescues in other Swiss cantons - many are happy to rehome cross country.
If you bring a dog in from any mediterranean country you should also have the Leishmanoisis test done. Most rescues will already have done this, but if not, any vet can do it here - it's a simple blood test. I had a full '
Mittelmeer' panel done for my mutts which screened for several diseases present in those areas. You can still bring a dog into Switzerland even if he/she tests positive, however it is important to find a local vet with experience of treating such diseases. Not all do, as they are not often seen here.
As to the issue of bringing the dog back to the US... Utter nonsense.

I adopted a dog in Hong Kong, and brought him (and his US born 'sister') back to the States without any problems. I'd be happy to speak to this woman on the subject if you think it would help (no Italian here, though... only English and Hoch Deutsch I'm afraid.)
It is common for a rescue to want to know where their dog is at all times, though. I have signed clauses saying I will inform the rescue of any move. This is because, should you at anytime be unable to care for the dog, a good rescue requires you to return the dog to them.
I can understand concerns about not having a garden though. Most dogs do need the space to run, play and just hang out in a secure outside environment. And, dog ownership is more difficult without immediate access to a garden. (I've been through a lot of 2AM emergency garden visits...) For some dogs, a large garden is an absolute necessity. Some do well pottering around a small garden, some are happy without one as long as they get their four walks a day. It really is down to the temperament of the individual dog. BUT, that said, a dedicated owner can often find a way to meet a dog's needs, even if the set-up is less than what some would consider perfect. You should think carefully about the rescue's concerns, they are valid - then work up a plan as to how you could address those concerns given your circumstances. This rescue may not budge, but at least you've given it your best shot. And another rescue just may be impressed that you've thought this through.
As for an older dog... Most of my dogs were adopted as oldies. Yes, I know my time with them may be short, but that just makes it all the sweeter. Some of my seniors are pretty wobbly, one of the oldest is fit as a fiddle and quite active. It's a personal choice, obviously, The important thing is to think honestly about what you want in a dog, what you can provide, and then look for the dog who meets your needs (and you his). It is often a case of something just 'clicking' when you find your new friend. But, oldies have their own special charm.
I'm sorry you've had this bad experience, but please don't let it put you off rescue. Go speak with other Tierheime, take a look at the Tierschutz database, look at the Italian sites. Don't lose heart - you will find your four legged friend.