Since there is a new release version (6.0.1125) of Avast that helps prevent crashes, I would upgrade via the GUI and see if this helps.
Are you connected to the Internet via WiFi, wired, cellular?
Is your firewall (FW) settings set to public? Do you see any evidence of intrusion? Is your router locked with a strong password? If you go to Network Connection (? correct wording for Win7) > scan your machine for another connection > do you see another user who is unlocked that may be accessing your machine and do you see an unknown IP address in your FW log?
May I ask why you leave your machine on while not attended to for such long periods of time? Have you tested to see what happens with your machine if you prevent it from going to sleep and actually shut it off when you are not using it?
How many programs do you actually have starting up at boot-up? Are they all necessary or can you "disable" some of them to speed things up at boot-up?
Since you leave your machine on all the time, do you ever clean your temporary Internet files with something like CCleaner?
Forgive all the questions, but I'm trying to look at the big picture.