A galaxy isn't a planet. Planets themselves are rarely perfect spheres. Earth is actually an oblate spheroid. Solar systems also exhibit planar geometry. They all spin. The angular momentum is enough to overcome the gravitational pull. If that momentum was not enough to overcome gravity, all that mass would collapse into a spherical object. Depending on how much mass would determine what kind of celestial body it is.

Not all galaxies are flat, and the stars in galaxies are sperated by enormous distances.

It's the difference between Saturn and its rings. Saturn the plantet collapsed into a spheroid. The rings are pieces of stuff, not solid. Of all the stuff in the rings were thrown together into one solid clump, it would be a spheroid, too.

Yep, as are solar systems. But when they collapse in to single objects, those objects end up spherical once they're about a third the size of our moon (or even less than that if they're icy/gaseous).

In this case, it's all to do with the conversation of angular momentum! This is also why the solar system is in a plane.

Basically, during formation, the cloud of gas forming the galaxy is spinning super fast, and the difference in rotational forces flattens the cloud closer to a plane.