I read here that galaxies become disks if there is a lot of gas in them, since their angular momentum is conserved while their energy decreases due to collisions of the gas particle. I have two questions about this:
Why is the disk the configuration that has low energy and high momentum? Can't there be a spherical shape that meets the same conditions?
If stars are formed when gas collapses under by gravity, why doesn't the gas create a disk shape, but instead creates a spherical shape?
Disk galaxies form as a consequence of the existence of a non-vanishing tidal field at the time the gravitational collapse of the galaxy decouples it from expansion. This is a good reference if you want to read the details at early epochs of this process. But, as you mention, it boils down to conservation of angular momentum. These galaxies are said to be supported by rotation, in the sense that the rotation velocity dominates the random motion, in other words the number $V_{\rm circ}/\sigma$ is large.
The problem is that kinematic cold disks are in general very unstable, and even small perturbation over time can destroy them, ending up in galaxies with more spherical shapes. Evolution in a hierarchical universe, such as ours, will naturally lead to young disk galaxies and old elliptical galaxies. Although this is still a matter of debate.
This argument should also give you an idea as to why there are no disk stars: at the time of formation, material indeed can be accreated in the shape of a disk, but over time, all the instabilities generated by the formation of the star would relax the system to a spherical shape
Why are stars round whereas galaxies are disks?
Nice question which is a approximation only of what happens in reality. Our earth is not a sphere but a sphere with flattens poles. I suppose that one can find stars for which this holds too. On the other side galaxies could be beside others lenticular galaxies or eliptical galaxies.
For me it looks like there is a coincidence between the mass density and the angular velocity of the observed object. Earth and stars are more solid while lenticular galaxies are at the other end of the classification. Knowing the radial velocity of a galaxy the flatness of the galaxy seems to be a good criterion for finding out the contained masses of this galaxy. Perhaps in detail it would be possible to refine the process by taking the thickness in several radial distances of the galaxy and get a good idea about the massiveness of galaxies.
To make my thoughts clearer
as well as we could conclude from the shape of the rotational paraboloid of a liquid in a rotating buckets (with a - for all observed liquids - constant angular velocity) about the viscosity OR for the same liquid on different planets about the different gravitational potentials
as well we can use the model of a liquid and instead of intermolecular forces calculate the gravitational forces between the stars in a galaxy.
Hope someone will show that this is what astronomers are doing.
The post you're referred to explains reasonably well why spiral galaxies are discs. So why are stars not disc like. The simple answer is: a disc-like object cannot be a star.
But what is a star? A star is an object that at its core has the right conditions for thermo-nuclear reactions, in particular temperatures $T\sim10^8K$. There are gaseous disc that reach extreme temperatures, but these are accretion discs around supermassive black holes, i.e. they are not gravitationally self-supported.
The temperature in a self-gravitating gaseous disc of comparable mass and size as a star is necessarily much lower. Moreover, any viscosity (which may be due to magneto-hydrodynamical or gravitational processes within the disc) inevitably promotes angular-momentum transport outwards and mass transport inwards, resulting in the formation of a star at the centre.
The stars were roughly disk shaped as the nebula that formed them was collapsing. The matter in galaxies is much more sparse. The thing that keeps them disk shaped is the centrifugal force as the stars and other bits orbit around each other and around the galaxy, and continually miss each other, otherwise, the galaxy would collapse into a big black hole. There is a black hole in the middle made up of stars that didn't miss each other.😕
