Ive noticed two expressions for 'working for'. Firstly, '...dla dyrektora', and secondly, as in tbe current example, '...w policji'. Presumably 'dla' is used for working with people and 'w(e)' is used for working for businesses and institutions? Or is there some other reason? Thanks.
Well, with people you can only work "for" them, with institutions, both versions will work. Sometimes we could say that "dla" and "w" mean the same, but it's also possible that you work 'for a bank' but not 'in a bank' (you are not exactly on the bank's payroll, you're not their employee).
Most of them are gendered. Generally the feminist tendency is usually different in Poland than in the English-speaking countries: most feminists seem to want more gendered words, they create "psycholożka" from "psycholog" (psychologist) or "architektka" from "architekt". Some new words enter the language quite easily, some are mocked as absurd-sounding and unnecessary - after all those professions have used the masculine word for both genders for a long time. On the other hand, there are women (and some of them may consider themselves feminists) who think that the masculine word is good to describe a woman, even if there is a well-established feminine word. For example I can easily imagine a woman saying "Jestem nauczycielem" for "I am a teacher", although the feminine word "nauczycielka" is a perfectly natural and common one. That depends on the specific word and the user, of course. A woman calling herself "policjant" - yeah, I can imagine that.
A gender neutral term... well, maybe you could go with "oficer policji", because "oficerka" is rather a type of a tall (?) shoe, although I guess "oficerka" could also be used for a female police officer nowadays. Anyway, "oficer policji" seems a lot less common than its English equivalent.
The problem is that those prepositions' usage is oftentimes multi-faceted and not always bound to one meaning precisely. With many prepositions, what they mean depends on the context. For the beginning, it's better to write down all the prepositions' meanings and also note which case they take in which context.
The English language's problem is that it is vague about some words' meanings, including “with”: A more literal meaning of “w” in this sentence would be “at”—the policeman works at the police commission, so to say. If we chose “z” instead of “w”, the sentence would apparently be: »Policjant pracuje z policją.«
I hope this helps at least a bit, and also do I hope that someone who knows better corrects my misconceptions, although I think I missed all of them.