Google could find a better explainer than what I can do...but I will try.

Yes, the universe is expanding in all directions. The "flat" describes not the shape of the universe but the curvature of space within the universe. In flat space, two photons traveling beside each other will travel in parallel forever (unless they run into something, obviously). In space with a positive curvature, they will part ways for some time (maybe a very long time) but will eventually come back to each other. If space has negative curvature, the photons will never see each other again.

For the most part, physicists want to know the curvature of space because the end of the universe depends on it. In a nutshell, a positive-space universe will eventually fall back in on itself. A flat-space universe will continue to expand forever, though its rate of expansion will tend toward zero as time approaches infinity. The negative-space universe will expand forever, accelerating as it goes. This is all based on the classic view of the universe; I'd have to think that the discovery of dark energy changes things, since it is causing our rate of expansion to accelerate. Exactly what dark energy (or dark matter) is is something of a mystery.