You would not always know if a virus got in or not, not if you were browsing the web and one of the sites you visited was hacked. No, it's would not have to be a dodgy site. Script-blockers would help to prevent an issue here yes, but that is one example.
Indeed, except that it would have to be a dodgy site. Why would a legitimate site try and exploit your browser?

I just browsed back through some Firefox security vulnerabilities for example, and the most recent one that mentions "remote code execution" is
https://www.mozilla.org/en-US/security/advisories/mfsa2016-92/ (Nov 16)
Digging into the tracker notes it required a specially crafted SVG animation, but would then run arbitrary code. The browser would then crash. This is pretty typical for a browser exploit - it requires a specific, crafted element on a site, i.e. a dodgy site.
Very scary all the same though - keep your browsers up to date people!
You can do what you like, but essentially if you are running no antivirus AND push family members away from an Antivirus who may not be technically savy, then you are asking for trouble.
I totally agree. It's very easy for even fairly competent PC users to run bad executables when trying to install something, for example.