Nearby our house is a transformer for underground power lines. It sort of looks like this: 
When I take a telephone pickup coil, and amplify I hear the 60hz hum through a speaker. I am wondering if there is enough magnetic field around this to pick up some power and light a LED or some low-power-draw device. If so, how would I do this, would I just hook a coil to an LED? Do these transformers leak magnetic fields in enough quantity to pick up anything?
Your question is a jumble of different issues. I will attempt to wade through them:
Can a transformer absorb power: Yes. It can both absorb power (which is why they get warm), and also not pass some types of electrical power through. But this is probably not what you really want to know.
Telephone Pickup Coil and 60 Hz: The 60 Hz you hear is likely unrelated to the transformer near your house. 60 Hz noise is all over (50 Hz if you're in some countries), and can easily be picked up by substandard electronics-- like your telephone pickup coil. There are lots of power lines around, inside and outside your home, and they all radiate 60 Hz.
Can you pick up that 60Hz noise and do something useful: Probably not. Ok, yes, in theory you could. But the amount that you would pick up is so small that it isn't very useful. Mythbusters did a show on this concept.
Do Transformers Leak Magnetic Fields: Yes. Some transformers are better than others. You could pick it up and do something useful with that energy, but only if you were very close to the transformer (almost touching the xfmr).
