ZA has form for this seemingly suffering from digital amnesia even when the avast.setup is allowed, updates don't work.
Find the entry for avast.setup in ZA and if it is allowed, delete the entry and do a manual update, that should force it to ask again, allow it and have it remember the decisions, etc.
Edit: Having looked at the log it doesn't appear that it is being blocked, but failing because of:
19:25:07 min/gen Error:Invalid file signature. Setup will terminate.
Try a repair of avast. Add Remove programs, select 'avast! Anti-Virus,' click the Change/Remove button and scroll down to Repair, click next and follow. You may need to reboot after the repair.
If that doesn't work it may require a download of the full virus database, but will cross that bridge when we get to it.