ashMaiSv.exe - Internet Mail Provider
ashServ.exe - Windows Service. Manages and implements avast! antivirus services for this computer. This includes the resident protection, the virus chest and the scheduler
ashSimpl.exe - Simple User Interface (skinned application)
aswUpdSv.exe - Windows Service. avast! iAVS4 Control Service, which provides automatic updating for the avast! antivirus.

All these files are listed on task manager. Just ashMailSv.exe and other file, named avast.setup must have the rights to access the Internet. None of them must have server rights (ie, could be accessed by the Internet).

Do you have a net card in your system (even if you do not have a net or LAN but just the card)? ZoneAlarm could ask for access rights if the LAN is not well-configured. I’m not sure, but ashServ.exe does not have to connect the Internet…

The IP addresses that are contacted must be found at the FAQ or help files of avast:

URL: http://www.asw.cz/iavs4pro
IP: 195.70.130.34

URL: http://www.avast.com/iavs4pro
IP: 64.246.6.135

URL: http://www.iavs.net/iavs4pro
IP: 207.44.156.15

URL: http://www.iavs.cz/iavs4pro
IP: 62.168.45.69

At last, you’re right, you could leave the resident protection on while you perform a scanning.