My overall goal is to be able to access public back-end databases via my own ad-hoc apps, in Python or JavaScript. This is data that is publicly available, but only via one-off searches.
More specifically, I'd like to access county property records, which are often a ArcGIS ESRI product.
For example, I'd like to access the every property's info attributes from http://cityview.baltimorecity.gov/cityview_D21/. I see in the code it uses ArcGIS and Data Dojo, though I do not know enough about these.
How would I begin accessing the script to call the whole database? Would this be a RESTful API? How would I know?
The best way to utilize the data you seek is via the server rest service. Looks like they have most of the data on that map available for use via https://geodata.baltimorecity.gov/egis/rest/services/. Depending on the capabilities they have enabled for the layer you can send queries etc. via esri's REST, JS, and Python APIs.