From my perspective, that odd lighting and tint is part of the aesthetic that makes it great. You can learn more about how the renderer works here:

http://fabiensanglard.net/quake2/quake2_software_renderer.ph...

You can create your own palette using the included pcx2pal: https://github.com/klaussilveira/qengine/blob/master/src/too...

After that, you might want to generate the colormap: https://github.com/klaussilveira/qengine/blob/master/src/too...

If you enjoy the code of Q2, but also would like more modern rendering, I highly recommend quake2xp from Barnes: https://www.moddb.com/mods/quake-2-xp

Yamagi is the gold standard for all Q2 engines: https://github.com/yquake2/yquake2

And some art help: http://maps.rcmd.org/tutorials/q2_palette_textures/