As an Avast user like yourself I don't have a specific list of avast versions for each OS. But that is going to differ for different users, a stable version would be hard to determine if it didn't effect all users.
In theory those with Windows 7 should be able to use the latest version of Avast, unless of course they finding problems with their system configuration/applications and a new version.
The problem on where to find the installer is also a problem, Avast only promote/provide downloads for the current avast version. You are looking at download sites that also have old versions of programs, such as file hippo or major geeks. Whilst some of those have supposed links for old versions, many just provide a link to a stub installer, this is a small downloaded file. That initiates the installation on-line and it is here that it fails, as it installs the latest version.
Windows 7 is next in line for MS to kill and how much support it too will receive from companies such as Avast, unlike XP I haven't seen any end of life/support for Windows 7 by avast.
My win7 system is a netbook:
Acer Aspire One, Win7 Starter (32bit) 10.1", 1024X600 screen, 2GB DDR3 RAM, Intel Atom N255 (1.5GHz dual core) CPU and 250GB HDD.
This is no power system and I use less and less each month/year and is probably not long for this life. I would say its biggest problem (excuse the pun) is its size, or more the resolution. The latest avast versions the UI is absolutely massive as is the installation screen.
What version you use is determined by many things, but something everyone can do when installing avast, is to do a custom install, the minimal install is an option during installation. For me that is too minimal, but each person is different, just don't install everything, just what you need/want.