Is there any problems, or limits, in using the same API key for three copies of the same site - one each in development, staging, and production?
In addition to the "fair use" question, I'm also concerned about URL logistics. Will the CM server have problems if three different URLs are making requests using the same key?
Nope, there won't be a problem with using your same API key in your 3 environments.
Excellent. Thanks Ben!