Emeralds do not spawn in veins. When a chunk generates, 5-8 stone blocks of that chunk have a chance to turn into emerald ore. The stone blocks are usually scattered, so it's very rare to find emerald ore close together, like a vein. Emerald's found anywhere from layer 40-1, I believe.
Everything can be found under level 16, but I personally dig with my head (my y-coordinate) at 12. This way, I'm in prime diamond range, and, if I encounter a pool of lava, it's at the perfect level where I can just pour some water on it and walk over the obsidian.
Layer 14 is the best layer to mine at. It's a few blocks above lava, so you shouldn't have to worry about falling into a pit of lava. This is also the lowest layer for lapis to spawn at, so you're getting all ores, except maybe emerald, because that only spawns in the Extreme Hills biome.