How do I verify Quad9 is working and that I am benefiting from its features, especially encrypted DNS and DNSSEC?
Did you specifically install, enable, and configure those features? Also, what protocol for encrypted DNS did you set up (DNS over TLS, DNS over HTTPS)? Pointing your router's DNS setting at a different server doesn't automatically activate new protocols and features.
Do I need to use a client on each computer/device, or does configuring my router make it work for the entire network?
That depends, are all of your devices pulling DNS settings from DHCP, and is the advertised address that of the router itself? It will depend on the individual device. If you cannot change settings on a particular device, you could add firewall rules on the router to force all DNS traffic to use the router's DNS forwarder.
If using encrypted DNS and HTTPS (only), am I protected from snooping, including by my ISP?
Assuming you are only using those protocols, the ISP can still see the IP addresses you are connecting to, which can sometimes be enough to determine the website. Also, the TLS SNI header (unless using the encrypted SNI extension to TLS) can reveal the hostname of the website in plaintext (e.g. example.com).
Is it bad to use encrypted DNS with Tor as mentioned here: https://www.privacytools.io/providers/vpn/#info ("However you shouldn't use encrypted DNS with Tor.")?
The link you provided already contains some reasoning behind this, although most users are just using the Tor Browser and DNS isn't even used on the user's end (the Tor network will make DNS requests on your behalf when you make connection requests to Tor's proxy). So this would only matter in certain circumstances; I would say that page is slightly misleading.
