Dead labor is any capital used for production. It produces no value because without living labor to run it, it does not produce. Even if you were to perfectly automate a production line, there would still be some worker looking over the process -- that is the worker that is creating and preserving value in a commodity.
There are two kinds of division of labor: manufacture and social. Manufacturing division of labor is a division that arises out of necessity on the shop floor, a "necessary evil" in capitalist society, which also alienates labor, in order to keep commodity production as efficient as possible. Social division of labor, which is the idea that only certain people can do certain things, and it is used to control society. He uses the example of the family in Capital, Chapter 15, and also how we've separated agricultural labor from industrial labor, thus creating the schism between "town and country."
For Marx, since socialism seeks to do away with alienation and oppressive social control, it should be necessary to do away with these forms of labor division. In an era of socialism where things are still in the process of being automated, the manufacturing division of labor is one of the "birth stamps of capitalism" that would exist, until we got to a point where it wasn't useful anymore. Humans generally aren't into hardcore specialization, so doing away with the division of labor would help people seek what ever pursuits they desire. People will still have their specialties ("For making boots, I defer to the bootmaker") but they wouldn't be tied to that, as if it was their only work role. The social division of labor would be done away with, hopefully, with some immediacy; for example, Marx calls for the industrialization of agriculture in the Communist Manifesto, as an immediate task of revolutionaries.


