If i surfe an get infected, does avast see it right away?
I also noticed that a program called “spooler subsystem” had gotten access to the internet by my zonealarm without my knowledge, is that a virus?

If the spooler subsystem has gotten access through ZA without your knowledge than either you haven’t setup ZA as you wish or someone else granted permission. Spooler subsystem is for your printer/plotter.

If i surfe an get infected, does avast see it right away?
Depends on what settings you have in Avast and the way the infection spreads.

Ok, then the spooler subthings is my hp printer

I got a warning online and deleted it