I don't think it is an adjusting screw, more likely an air bleed jet or something. It should be screwed all the way in, but you could remove it and see if there is any sediment collecting in the recess. That might be why 'adjusting' has an effect on the running.
One other possible pointer to your running problem. I assume that when you took the carburetor apart you didn't change the gaskets? Your rough running symptoms sound more like air leaks to me, and by taking the carburetor apart you've probably disturbed the gasket material and it is no longer sealing correctly.
There is very little to adust or go wrong in the Weber 32TLF carburetor, so if the float and float pin are fine and all the jets are clear, I'd advise getting a carburetor rebuilt kit or even just replacing the gaskets (you might be able to get them from Fiat).
Either way, don't despair as I'm sure you haven't damaged the carburetor beyong repair.
Hope you get it fixed