There are several, and there may be even more most would require both outbound and inbound.
The AvEmUpdate.exe that you mentioned, checks for emergency updates outside of normal updates, if one is found I presume it would require inbound connections to install it.
AvastSvc.exe is the main service, controlling the various shields.
Also AvastUI.exe, ashUpd.exe there is one I recall from earlier versions, instup.exe (Installation & Update function). EDIT: instup.exe is in the avast\setup folder and I have noticed instup.dll (I don't know if this requires access) also in the setup folder.
I have never used the win7 firewall, I have always used a 3rd party firewall (much easier to control) until I got windows 10 and on that system I use the windows firewall.