And - as I know - in webmail Avast can't filter e-mail attachments with possible viruses.
Yes it does, if you save a attachment from webmail / download a file from the net / any in/outgoing data traffic ... it is checked by your antivirus realtime ptotection
Also all mail providers have spam/virus filter on there mail servers (i dont know of anyone that dont)
Examples:
Gmail use postini spam/virus filter with two AV engines from McAfee and Authentium
Yahoo use Symantec/Norton
Microsoft (outlook / hotmail /msn mail) use Trend Micro
Many others use ClamAV since it is free open source