Do I need to access them via a proxy page?
...how it appears that I cannot host the proxy on my ArcGIS server.
The proxy runs on your local web server, not on an Esri server or on the computer where ArcGIS Server is installed (unless your web server also hosts the ArcGIS Server instance).
...as such I will need to create a proxy page in order to consume their data. Is this correct thus far?
it appears that I can download and set up the proxy page on my ArcGIS server seeing how both my webmap and my ArcGIS Server instance is on the same server. Is that correct?