How many times a day or a week does Avast!4.x Home update its virus database or itself?
Depends ;D
Program updates could vary from two for a year (although there are a lot of beta updates).
Virus database updates could vary from daily to few for a week. You can see how often it is updated: http://www.avast.com/eng/vps_history.html
Here is described what the VPS version number means: http://forum.avast.com/index.php?topic=3878.0
There is no set schedule for VPS updates they are released as they are required, they average about 4 per week, but I have had 4 in a day in the past.
With the avast auto update, it checks every time you connect to the internet avast will check for updates if one is present it will be downloaded (depending on your Program Settings, Update (Basic) settings). If you remain connected it will check every four hours or a delay that you can set.