Why is our galaxy planar? Why isn't it spherical?
Because it has rotational kinetic energy -- it's spinning.
Though it's worth mentioning that the process by which it obtained that rotation is still relatively unclear. Our best theory at the moment is that it was due to tidal torques from other pre-collapse galaxies.
I think its more spherical than it looks. There's lots of dark matter in the halo.
That depends on what you're referring to. The dark matter distribution is indeed thought to be spherical (roughly), but the gas is clearly in a disk. The fact that the dark matter is non-interacting means that it wouldn't be able to organize itself into a disk structure despite having non-zero rotation.
Separate names with a comma.