This is a very broad question, we have a huge numbers of such characterization.
But part C of "Sketches of an elephant" contains most of those I know (especially C3). Basically all the notion introduced have such a "site characterization", i.e. a properties of a topos or a morphism is characterized by the existence of a Site description having some properties.
And I would like to add that Part C is in my opinion the better written and easiest to read part of the elephant.
From memory, you can find there at the very least conditions for:
- Atomic toposes, as sheaves for a topology where all non-empty sieve are covering (called an atomic topology).
- Coherent toposes by combining the fact that every covering is finitely generated with existence of finite limits in the site. ( it is a bit more subtle than that, I refer you to the elephant for the precise statement)
- A characterization of proper geometric morphisms. (but maybe not the simplest possible ? )
and a lots of other examples (open geometric morphisms, locally connected morphisms and so one...) but I do not remember all of them.
This dual frobenius property that you sate seem related to properness (maybe be not equivalent, but that is what it makes me think about...) you should have a look to the section of the elephant on properness.