1. Yes
2. Potsdam is a must, though it's more like a half-day trip. Dresden is an option for a full-day trip (not a must, but certainly worth visiting).
3. Possibly, depends on your travel preferences. If you're a slow walking, museum-visiting type of traveller then you'll have enough to do in Berlin alone.
4. Hotels are cheap - £50 gets you at least a 3* in a reasonable location and I've stayed in a 4* hotel close to the center for about the same price (though in late November).