taps
I've made quite a few of these tap handles for my self and friends. Yes the thread size is 3/8 x 16tpi.
As long as you use the threaded inserts no other hardware is necessary it just screws onto the top of the faucet. There are several kinds of threaded inserts you can use. I've used others but the brass ones with the notch for an installation tool seems to look best. If the insert is not screwed in straight the handle will wobble as you screw it on and the final product wont sit straight. It's more obvious when several are grouped together.
If you don't want to use the inserts then you can use the ferrules.
The ferrules are just for decoration and thread onto I think its called a hanger bolt. It's half wood screw and half threaded bolt (3/8x16).
Handle sizes vary, it usually depends on his kegerator and how it is set up.
Hope this helps any other questions let me know.