Learn Polish: Kobiety były nam życzliwe i przychylne pomysłowi. - The women were kind to us and favorable to the idea.

Both życzliwy (kind) and przychylny (favorable) are adjectives that mandate the Dative case for the person or thing receiving the attitude. The women are kind to us (nam) and favorable to the idea (pomysłowi).

Why is the pronoun nam placed before its adjective, but the noun pomysłowi placed after?

In Polish, short pronouns like nam usually come before the adjective that governs them, tucking them into the middle of the phrase. Full nouns like pomysłowi are heavier and carry more emphasis, so they typically follow the adjective.

Both adjectives end in -e (życzliwe, przychylne). What case and gender is this?

They are in the Nominative plural, specifically the non-virile (niemęskoosobowy) form, to agree with the feminine plural noun kobiety. If the subject were a group of men or a mixed-gender group, these would change to the virile forms życzliwi and przychylni.

Is there a rule for why the Dative form of pomysł is pomysłowi and not pomysłu?