I've tried everything I could find, asked on the IRC channel for help. I didn't get too far. After 5 hours of trying, I managed to get some low volume, distorted sound from my headphones. (My built-in speakers still work fine).
Any related details are included in the link pasted above.
The "sound" started coming form the headphones after I set every possible value in the alsamixer to the highest number and added the following code to the alsa-base.conf file :
options snd-pcsp index=-2 alias snd-card-0 snd-hda-intel alias sound-slot-0 snd-hda-intel options snd-hda-intel model=laptop options snd-hda-intel position_fix=1 enable=yes