Before you begin you need to define what you mean by 'building density'. Do you mean buildings per unit area or do you mean the area covered by building footprints per unit area? These are not the same thing. The former might be more appropriate if you are looking at (say) ownership but takes no account of the difference in size of building.
Consider two towns. One has a few huge buildings and the other has many small buildings. The land area covered in each case is identical but the number of buildings per unit area is radically different.
Next, how are you defining 'building'. Is a row of terraced houses, a single building or multiple buildings? If you say 'It must be multiple buildings because it has multiple owners and multiple front doors', what about flats/apartments? Also, you could equally argue that a terrace should be considered to be a single building because it was designed as a single unit by the architect.
To continue, many might consider building denisty in a downtown area to be higher than in a semi-suburban area. They may be right in literal terms but then again, many suburbs can have as much ground coverage as the downtown area but be considered less dense because they are lacking in the third dimension (building height). OK, that last point is legitimate but stretching things a tad. However, I'm trying to make it clear that mapping 'density' is not quite as straight forward as may first appear and that your definition of 'density' will change the way you do your analysis. On that note have a look at this video.
ONE way to approach for each of your questions are as follows (there are others and, like I say, much depends on your definition of density):
- Convert the building footprints to points and then calculate point density (this is like my first example above).
- Get the number of raster cells coded as buildings, multiply by the resolution and then divide by the total area of your raster (this will tell you nothing about clustering, auto correlation etc though).
You need to read up on spatial statistics for a full answer, but a trawlthrough the capabilities of Spatial Analys extension will be an excellent start.