I'm still relatively new to NGINX Proxy Manager myself, but I'll give your questions a shot. It doesn't matter how (or where) you host your proxy instance, what matters is that the requests can get to it so that it can forward them to the correct resources. So simple answer to question one is no you can host locally.

If you host it locally you need to make sure that you forward requests that come into your network on to the proxy to be routed correctly. This is where port forwarding comes into play. You'll need to set your router to take any requests that come in on port 80 or 443 (HTTP and HTTPS) and send those to your proxy.

As for question two do you need to purchase a domain. You can use a free domain name or you can pay for one that part doesn't matter. The domain isn't a technical requirement until you want to start hardening your instances with SSL. To get a cert you'll need a domain. But if you set up your port forwarding and a proxy you could send a request to some_subdomain.123.456.789.123:80 (your external IP) and the proxy server will take thar request and translate it to the local server mapped to some_subdomain.