Also, våning, it seems, is used in a different context than golv, for "floor", as in the former refers to the level of a building, and the latter, to the actual physical floor in a room. Would I be right in this?
Yes, we have two different words for this. golv is only the floor in a room, never 'floor' as in 'first floor, second floor', those are always våning.
en våning can actually sometimes be that too, but only an upscale version of it. So most apartments are just lägenheter, but the luxury ones can be våningar (you can also just call them lyxlägenheter).
It's the word samma that doesn't go together with definiteness. It's like things get definite enough from that word alone, so no more definiteness is needed.
In Sweden is it 'ground floor, first floor, second floor', or is it 'first floor, second floor, third floor'?
The translation says: we live at the same "story". This is the wrong English word. The correct word is "storey" i.e. level or floor. A story is what you write or tell, like "Little Red Riding Hood".