Can someone help me out here?

So bottom brackets come either in JIS or the italian standard?
That means the part that the crank is going to attach to?

So depending on the cranks I want, I choose a bottom bracket with the matching style?

What about the width of it? That has to do with the chain?

width usuall has to do with the frame and if it will fit flush with the BB area. At least I think so.

So I have to look at my frames specks, to find the width for the BB?

Looks like the rear hub width and the cog should also play a role in determining the width.
Isn't that gonna change the way the chain will run?

Square tapper bottom bracket spindles are either JIS or ISO but there are many different threading of bottom brackets, ISO, French, Italian etc etc.

It is advised that you use the bottom bracket that the company of the crankset suggest; both interface type(JIS/SO) and spindle length. You need the correct spindle length to have your chainline as straight as possible.

Width is determined by the frame, measure your frame's bottom bracket shell. You'd need a bottom bracket that is the same width as the shell. In your case, a Gazzetta, which use an ISO/English threaded bottom bracket, you'll need a 68x1XXmm bottom bracket. The latter number represents the length of the crankset, which again, determined by the crankset.

Reading Sheldon's article on bottom bracket might help a bit.