Update:
I replaced the light, it works fine. I had no idea that the bulbs has 2 filaments. It makes sense now that one mode wouldn't work.
As for the transmission, it kind of got fixed by itself, I guess by using it. I still went to see a mechanics yesterday to get a general inspection and he told me, as you said, that it's either the transmission cable or the transmission lever IIRC. He suggested that I change the lever for now, and that should fix the issue.
The thing is that I looked on rockauto.com and could not find the lever in question, is this the correct term?