How do integrated circuit design companies create their datasheets?
It depends very much on company culture, as The Photon implied.  In some companies the designer(s) is/are responsible for the datasheet.  In others it is an Applications Engineer.  I can only tell you my personal experience.  The take away is datasheet development really informal.

For chips I've been lead engineer for, I've written the datasheet myself using Word.  The first draft of the datasheet is often the first part of the Microarchitecture Specification (MAS) which is written early in the design process.  In big companies, as Null said, this is driven by the Systems Engineer.  Where I work, the designer is usually responsible (and takes on many of the roles of a systems engineer).  After the design is finalized, the chip is taped out, and I'm waiting for wafers, I will take the beginning of the MAS and then turn it into a preliminary datasheet.  All the text is written, and simulations are put in for all the plots.  This is passed around the group for feedback.

Then, the chips come back and the Product Engineer does bringup.  The Test Engineer starts measurements and I replace all the simulations with measured data.  I also replace the specifications table with real measured data (sometimes specs change after the chip comes back, you know!).

Then, when I deem it complete, I send it to our technical writer (we only have a few where I work) and he or she puts it in the standard form.

As The Photon said, during the design process a lot of ad hoc procedures are used.  I used to use Excel for keeping track of pads and signals, now I use Google Docs.  The actual DESIGN data, however, is kept in specialized databases accessed by expensive CAD tools with version control and so on.