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On reading the documentation of the MySQL join commands, it looks like all the joins are analogous to , by simply finding the Cartesian product and then selecting from that result.
Is this an accurate assumption?
Should I instead write my own sub-queries and select from those?

A: 

Logically you are correct, joins are Cartesian products that are filtered; but practically the DBMS query processor is smarter than that.

You would need to look at a show plan or explain to see what it is doing under the covers.

Assume that the Query Manager knows more than you do about how to create a fast and solid query plan, don't pre-optimize.

Rawheiser