This already tells me that your router is configured to use Pi-hole as its upstream DNS server, which is also confirmed by your debug log:
*** [ DIAGNOSING ]: Discovering active DHCP servers (takes 10 seconds)
Scanning all your interfaces for DHCP servers
* Received 548 bytes from eth0:10.0.0.1
Offered IP address: 10.0.0.31
DHCP options:
Message type: DHCPOFFER (2)
router: 10.0.0.1
dns-server: 10.0.0.1
This is a valid configuration:
DNS traffic in your network will be filtered by Pi-hole, but you won't be able to attribute DNS requests to individual clients, as all requests originate from your router.
Conditional Forwarding has no bearing on this - it would only allow you to associate hostnames as known by your router to IP addresses as seen by Pi-hole. And Pi-hole currently just sees your router's IP.
If you want to change that, you'd have to configure your router to distribute Pi-hole as local DNS server via DHCP instead of using it as upstream.
You'd have to consult your router's documentation for configuration details or whether that option is supported at all, respectively.
Yes, though it may not be labeled that way in your router.
The important part is that your router distributes a DNS server via its DHCP server.
Commonly, you'll find a respective DNS option as part of a LAN / DHCP setting in your router somewhere. You router's documentation should have the details (if it supports it - there are routers out there that don't).