Absolutely right, any parts removed from your car are your property unless Fiat decide to honour a warranty claim. Just make the Service Manager absolutely clear that they must not bin them. An honourable dealer should return them to you any way with a nod and a wink if independent scrutiny is appropriate.

You should always get your old parts back, even on a service, if you ask for them and tell them before and is a condition of them doing the work.

On my 500X services I've asked for air and pollen filters and spark plugs back. I've also has the remaining litre of new oil returned to me. One of the spark plugs has already gone into the boot of the car for emergency use. The filters reside in my garage as pattern/templates for when I start servicing the car out of warranty. I can check the new filters I've ordered for correctness before I start the service. Nothing worse than starting a job then only to find out the parts don't fit.

You are quite within your rights to have the RAC, AA or another independent person to inspect the parts and problem whilst the garage is doing the work, if you can't be their yourself. Of course this would be at your cost, even if it is a fully warrantable claim. Such a person would also grab some pictures for their own purposes in support of any decision they come to. What's more despite the extra cost if it turns out that the claim is not warrantable then at least you know you have not been ripped off.

I would suggest taking it to any fiat dealer when you are able to demonstrate the problem. They should then inspect/dismantle it to locate fault.
I have experienced the 'no fault found charge' on one of my fiats, and that is why I think you need to demonstrate the fault to them first.

Warranty fix up to 3 years should be no problem providing your service record is uptodate .