1. Is it possible my new server's IP is blocked by them
Doubtful, but their developer is a pretty cool guy, feel free to just outright ask him if your server is blacklisted (I can't see why they'd have you blacklisted; a large chance is that your host is blocking outside connections to file_get_contents).
2. When I look on that web site, it appears I may need an "API Key?" Thing is, this script was working before I moved to the new server.
Stop Forum Spam requires an API Key to send data to their database; a key is not required to send a remote query with file_get_contents, but file_get_contents needs to be able to contact to "the outside world" (some webhosts restrict remote content).