A door register? Well, a register of people who live behind the same door. In other words, a family register. (Think of "door" as shorthand for a house in this case.) Family registers are actually pretty interesting. Look up "koseki" on Google and you're sure to find many fascinating histories.

The reading for 戸 is an exception. Instead of と it's こ. Just think about how the person who runs this census doesn't care about number of toes. They care about number of 子 (こ).