> How can I detect if my router is backdoored, or being used as a residential proxy?

Aside from the obvious smoke tests (are settings changing without your knowledge? Does your router expose access logs you can check?), I'm not sure there's any general purpose way to check, but 2 things you can do are:

1. search for your router's model number to see if it's known to be vulnerable, and replace it with a brand-new reputable one if so (and don't buy it from Amazon).

2. There are vendors out there selling "residential proxy IP databases", (e.g., [1]) no idea how good they are, but if you have a stable public IP address you could check whether you're on that.

[1] https://ipinfo.io/data/residential-proxy