If i stand directly below a big, dark room, how many blocks may there be between me and the room for mobs to spawn?
from the WIKI
Mobs spawn naturally within a 15x15 chunk (240x240 block) area around the player. When there are multiple players, mobs can spawn within this distance of any of them. However, mobs that move farther than 128 blocks from any player will immediately despawn (see Despawning), so the mob spawning area is effectively limited to spheres with a radius of 128 blocks, centered at each player.
Now, keep this in mind.
A mob that has had no player within 32 blocks of it for more than 30 seconds has a 1 in 800 chance of despawning on each game tick (1/20th of a second). That means that such a mob will survive for 40 seconds, on average, from the time it spawned or was last within 32 blocks of a player. Equivalently, the half-life of monsters not within 32 blocks of a player is 27.73 seconds.
This means, if you are more than 32 blocks away from your mob spawn area, you will need to use a flooding or piston based spawn to move the mobs to your kill area.
I have a rather small high altitude spawner that floods every 30 seconds or so that works wonderfully for its size, Mine uses a redstone piston clock timer to power the flood gates.