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The Sun currently has a stellar class of G2V. However, since its birth, the Sun has brightened significantly. What would the Sun's class have been when it was born ~4.6 gyr ago?

Spectral type is determined by (and is an indicator of) photospheric temperature and surface gravity. The classification of G2V applies to a star with a photospheric temperature of about 5800K and a surface gravity of about 100-1000 m/s$^2$.

The Sun's temperature and gravity have not changed greatly since it first appeared on the "zero age main sequence" - it has got a little bigger and more luminous and a little bit hotter (by about 170 K- Schroder & Smith 2008) - so maybe it was a G3V-G4V star at the zero age main sequence (Mamajek 2018).

However, there were much bigger changes during the 20 million years or so before the Sun reached the ZAMS. It was cooler and briefly much larger. This would be reflected in a spectral classification that was something like K5IV- M0IV; indicating a subgiant-like gravity and a temperature of about 4000K. As it aged, it would then shrink and become hotter, passing through the G5V-K5V spectral types before reaching the main sequence.

If you can synthesise these two diagrams in your head, you can see what I mean.

Pre main sequence evolution in the Hertzsprung-Russell diagram

Spectral types on the Hertzsprung-Russell diagram

Mass is one of the major factors that determine the spectral class of a star when it is born.

During a star's protostar stages, a spectral class determines the length of its protostar stage and other factors such as its lifetime. Even though a spectral class is not physically assigned to them, the life stages that they undergo are essentially determined by their spectral class at birth.

So even though protostars don't have a spectral class determined by scientists, the star that it ends up becoming in its main sequence is really what its spectral class is.

So in this case, the Sun in it's protostar stages still was on the G2V route - it was taking the same (or at least similar) life cycles as stars in the G2V spectral class.

Hope that helps!

To the 2nd part:
Check out this website, especially from page 18: http://www.astro.umass.edu/~myun/teaching/a100_old/Astro100Mar25.pdf. There's a lot of equations on there.

There's also more information here on the luminosity-mass relationship: https://www.e-education.psu.edu/astro801/content/l7_p3.html

On a side note, looks like someone asked about the luminosity+temperature and mass relationship on here as well: https://physics.stackexchange.com/questions/12589/are-there-formulae-for-calculating-stellar-luminosity-and-effective-temperature

If I find more information, I'll comment it or edit it onto my answer. Hope that helps!

It is true that on the main sequence, where most stars spend most of their lifetimes, mass is the main determining factor of all stellar parameters, with metallicity and angular momentum distant second-most important factors.

However when a star is born, from the moment it becomes detectable, it moves through several stages of structural evolution, which surely do not classify as the stellar class that it later has on the main sequence.

The evolutionary stages of this pre-main-sequence (PMS) star can be described in the astronomer's favourite diagram - the HR-diagram - via the Hayashi track.
The Hayashi track is for any star of given mass the sequence of colour-luminosity values that it attains during its PMS evolution.

As colour gives the stellar class, there is an evolution in stellar class for all stars, even on the main sequence.

However

since its birth, the Sun has brightened significantly.

is incorrect. Once the sun arrived at the main sequence, it still brightened, yes, but that means there was an increase in Luminosity. As the wiki page on the sun's evolution clarifies in the figure seen below, the temperature however didn't change much during that time, therefore also not the stellar class.

Summarizing
During the pre-main-sequence of a star the stellar class evolves significantly, on the main sequence it doesn't.