I literally laughed out loud when I saw "Is this doable in 4-6 days?"
To put it nicely, no this is not possible. It's completely unrealistic (and a bad idea) to attempt it.
To "hit" (as you put it) Rome + Florence + (Tuscany or Venice) would require more than a week, and that's at a punishing pace (I do that sometimes).
4 days? Pick one place. You could do Rome with a day-trip side-trip (I'd pick Florence if it must be someplace on your list).
Have 6 days? One, maybe two places. But no way you can see your list in the time you say you have.
I'm guessing you've never been to Italy before (probably never been to Europe, too). If I were you, I would do one or both of the following: extend your trip so you can see more of your Italy list, or make your list shorter (a lot shorter).