1) Blizzard games from my experience just play a lot better in a window, rather than full screen. Didn't even try D3 full screen, but I know this is true for D2, WC3, and WoW.
2) I'm a multitasker and always watching my Twitter, IM clients, and sometimes switch to a browser to do a quick search (only when I'm in town).
I'm suggesting playing in a window due to I've yet to have a cursor disappearing problem. Could also try playing in a window with the game taking up majority of your screen, so it's somewhat full screen.
Oh and I highly doubt that the physical mouse hardware that you have will have any impact on if the mouse disappears or not. Saw some mention the brand mouse they used. It's a software issue, in this case being a D3 coding bug, that is the problem. Not the actual mouse.