No you don't have to be on-line, in fact it is possibly better that you aren't (less other activity for avast to have to scan).
Yes you do need to be on-line to update, there is no other way to directly 'download' the virus signature or program updates.
The avast updates will automatically check when you establish an internet connection (they don't establish a connection). With the default settings if there is a VPS (virus signatures) update available the incremental update will be automatically downloaded; if there is a Program update available because this is likely to be much larger it will 'Ask' by way of a notification pop-up if you want to download it now.
Or is there something not written in this last question, e.g. a system that isn't connected to the internet ?
