These cases are extremely rare and it's very unlikely that updated and well maintained websites get hacked. If you think any antivirus will protect you from that, it won't. Antivirus software can barely protect you from phishing websites, let alone hacked websites.
The only thing that can protect you in events like this are maintained and updated web browser, DNS with security in mind (such as DNS0 or Quad9), and ad blocker with comprehensive list like HaGeZi. I hope I don't have to mention common sense.
Antivirus software isn't as quickly updated as DNS or ad blocking filters and for that reasons you can't expect them to catch malicious website which typically last an hour or so before they are taken down. Again, this is why phishing is so efficient; antivirus companies just can't keep up with it. By the time antivirus company adds phishing domain into their data base, website is already long gone.
Extremely rare cases. You're more likely to win the lottery than to have malware in the background without your knowledge.
Beside, sooner or later you'd notice someone got into one of your accounts or other signs that something just isn't right.
No one said common sense is enough. It is 80% of your protection though, if not more.
Common sense is the one that will tell you not to enter your credit card details after taking a survey for free iPhone, not an antivirus software. It will also say that you aren't related to any Nigerian prince so you shouldn't give him your data when you get a mail. And that you don't have a banking account in Las Vegas filled with millions that you forgot about.
Without common sense, there's no life.
Extremely, extremely rare. Almost non-existent.