CouchPotato and Sonarr are both working from my PC. I did a copy / paste with the API Keys from these apps and used those for nzb360. When I try to use them, I am prompted to login which I take to mean the api key isn't working.

Any suggestions?
Login to what exactly? If you're seeing results in search for Dog and PF, your API key is working correctly.

If you're referring to logging in to the website to view details, the API key doesn't work for that. You'll still need to login (once) for that.
I am being prompted to login into Dog and PFMonkey from NZB360. I assumed it would as Couchpotato & Sonarr work. If the API key is being used, a login is not required.
For the detail page in search? The API key is to grab content. If you're seeing content, the API key is working fine. If you're referring to the details page when you tap on an item, you have to login because this is the web detail view of that item. It's just a handy way to find out additional info about an item that the API doesn't support.