1. Quite simply, does the virus-protection itself work without this add-on? Or is it just an extra that warns you about sites. etc.?

Yes, Avast works without the addon.

  1. On that subject, what does “this site has a bad reputation” mean?

Avast calculates taking into account votes by the community and various security sources for example blocking known phishing sites. I am not sure how the algorithm works, how much community votes or other sources weigh, but basically don’t visit unless you know what you are doing. I believe a website could be rated “bad” based on questionable content and not necessarily on security, but don’t quote me on that b/c a lot of times questionable content and bad security go hand and hand.

  1. What does it mean when it says that 3 companies (or whatever) tried to track me?

I believe the number is on whatever url you are on. Almost every site tries to track you and use analytics in multiple ways so it is no biggie.

If you have problems with the Avast addon you can alternatively use
For Firefox & Chrome : My Wot addon + Adblock Plus addon (subscribe to Easy Privacy Filter, have “allow non some intrusive” option checked off) to get about the same effect. I prefer this setup over the Avast addon myself.