For our highly unstructured, untyped, non relational artefacts? It'd be fairly impossible to use it as a datastore in this particular case, and regardless, it would provide little to no speed increase over our current application, as the limiting factor is the CPU cost of the map function.
Maybe you should consider transforming the data to structure it a bit. Have a unique identifier per object and arrays for each seen characteristic with (value, id), and it's reverse index. Then decompose the processing of each object to sub-problems matching n-way the characteristics. It doesn't have to be in SQL, though. I'd try a columnar RDBMS. YMMV
I have extremely strong doubts about that. Everything can be modelled in modern databases, and SQL is probably much more powerful than you understand here.
Execution time, after a point, is less important than development time. NoSQL is often faster for development and refactoring, because the schema is easier to change in code than in the database.
Also there's the scale issue. Append-only is helpful. Why have SQL if you can't use all the features?