If it's a tight clump it's a clumper, if it's spread out all over the place it's a runner.
Bamboo is just big grass so any lawn fertilizer works well, once in the spring and once in the fall. For the spring feeding find something that encourages top growth, in the fall something that encourages the root system, of course like any plant, it responds well with a top dressing of compost, manure, or any rich organic material.
Division is easy, just try to dig a few culms with as much root mass as possible. From my experience here in the Northeast I have found that the best time to dig and divide is as soon as the ground is thawed. this way divisions will have all summer to produce new roots and rhizomes which will ensure survivability for the winter.
The American Bamboo Society is a good place for tons of info, as well as the Garden Web. Any chance of posting any pictures of it?
Good luck.