Is there a database of disease prevalence? I want to collect data on how common different diseases are in different populations (or, at least, globally). Ideally, it would give me a way of querying the database with a disease name, and would return the number of cases per N population. 

I have found FIDD, but it seems to be no longer maintained. Are there any actively maintained, similar tools that have the information I am after?
For rare diseases, you can use Orphanet. You can download an xml file with prevalence from orphadata.

There is an OWL version of Orphanet called ORDO, you can browse it on the EBI Ontology Lookup Service. See for example Arrhythmogenic right ventricular cardiomyopathy and point prevalence relationships in the lower right box. There are a variety of programmatic ways to access this, including the OLS web API, SPARQL queries, or downloading the OWL and using an ontology API.