After 18 months of ownership and around 18k miles my start-stop is now pretty reliable. As far as I know I’m on the original battery and in the past it has needed to be charged at home a couple of times - not because it wouldn’t start, just because s-s didn’t work. The battery must be at least 80% charged for s-s to work.
A couple of weeks ago I did a 350 mile round trip in one day, and since then it’s been perfect. Before that it was very hit and miss, stopping for a few seconds if at all or just giving the ‘unavailable’ message. I put it down to a good long run clearing out the dpf etc and charging the battery. We’ll see how long that lasts.
The Giulietta is well known for not tolerating discharged batteries - there are many threads covering it - so I would either charge it or replace it before going looking for other faulty components.
The HDC/hill holder/steering errors all come up when the battery has been disconnected, and there is a procedure in the manual to cater for this, including resetting the window switch stops.