Re 2. Did you 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 need to be on-line to do this.
Re 3. I don't know how the Push updates work, obviously not using the standard update process avast.setup otherwise that wouldn't work either.
Does a manual update work, right click the avast icon, select Updating iAVS?
Re 4. Which processes, does that include avast.setup?