Is 壁橱 (as opposed to 衣柜) only a colloquial word for "closet"? I apologize if this is very elementary: I've never heard of closet referred to as 壁橱， only as 衣柜， but the direct translation doesn't seem appropriate. I've also heard 衣柜 used to reference cupboard as well. Is this for colloquial reasons?
壁橱 can refer to a small closet, a wardrobe embedded in a wall, a cupboard hanging on a wall, a chest placed against a wall or something else. As the characters go (壁=wall, 橱=case), it can be a container for anything not limited to clothes. Note, as an imported concept, 'walk-in closet' is not translated into 壁橱 but something along the line of 储衣间 (clothes storage room) or 更衣室 (dressing room).

衣柜 (衣=clothes, 柜=cabinet) means a wardrobe, either built-in in the wall or a standalone piece of furniture. A general-purpose cupboard should never be called 衣柜.

In Chinese, the distinction between built-in in the wall vs. standalone furniture is not made by first-class words or affixes. Both 壁橱 and 衣柜 can refer to either a closet or something else. 

For the similarities/differences between 橱 and 柜, please refer to @Question Overflow's answer here.