Addressing the last part of your post
Why are the names of subshells based on completely random letters (s,
p, d and f)? I tried searching for some source on the internet but it
simply dwells deeper into their shapes and planes and graphs rather
than explaining their names.
These letters are not random and as you can see all over the internet (e.g., Wikipedia) they stand for sharp, principal, diffuse and fundamental. After that the story ends without telling what is sharp, principal, etc. What are those adjective for and what do they describe? These letters described the nature of the lines in an atomic spectrum and this system was not universal (Taken from "A Report on Series Spectra by A Fowler).
Basically, long before the concept of orbitals originated, people were fascinated by the spectrum of alkali metals. They have a rich spectrum if you excite them under an electric arc. When you view the arc through a spectroscope (=a device that separates wavelengths), you see a number of lines these letters described those lines as they appeared to the viewer! Pure visual labels.
Now you don't see fundamental there because the fundamental series was discovered in the infrared region. So no visual label, it is just a label. Nothing fundamental about it.
Now, spectroscopists, mathematicians, physicists wanted to find out a mathemtical relation among the wavelengths of those lines. They came up with mathematical series that described the wavelengths as terms of a series. When quantum mechanics was developed this nomenclature was retained since they were able to assign the electronic transitions in orbitals corresponding to those wavelengths. I will have check original papers from 1920s to see who first assigned them.
P.S. Saw an interesting comment below the original post. I want to correct it since some student later on start to spread this reasoning in future on other websites.
You may remember the order of the subshells spdf by the following
reasoning : s is the first letter of "spherical", because the s
orbitals have a spherical symmetry. p is the first letter of "plane",
meaning that the p orbital has a planar symmetry. d is the first
letter of "different", because these orbitals have different symmetry
surfaces. Other orbitals are defined by alphabetic order after "d",
avoiding the letter e which is used for exponential. Historically
speaking, the letters s,p,d,f were named after other criterias, which
are harder to explain.
I am afraid this is completely imaginary and there is no historical work which justifies the choice of the names orbitals using the above reasoning. The letters do not stand for spherical, plane, and different. The letter f was not chosen to avoid "exponential".