From the bottle to the glass
From the bottle to the glass would be "şişeden bardağa"
So would the correct translation for "I am pouring the beer from the bottle into the glass" be:
Ben birası şişeden bardağa döküyorum
And would "Can you pour me a beer" be
Sen bana bir bira dökebilir misin?
Was just getting a beer from the fridge and practicing in my head so want to know if it's correct :)
We would use "koymak" in that case: Bana bir bira koyabilir misin?
With "dökmek", it would sound like you want them to spill the beer on you, like, on your clothes and everything. That would be wasting the beer.
Ok, a bit like the way the Spanish "Put" you a beer as well.
And now i know what to say if I want beer poured all over me as well. :)
In your first sentence, "the beer" should be birayı.
(You wrote birası, which is "his beer", in the non-accusative.)