Learn Italian: Abbiamo fatto dipingere la casa all'imbianchino.

This is the main rule for the causative construction when there is a direct object! Because the infinitive dipingere (to paint) already has a direct object receiving the action (la casa), the person performing the action must become an indirect object. That is why we use the preposition a (combined with l' to make all') before imbianchino.

In English we say "had the painter paint." Why are fatto and dipingere right next to each other in Italian?

In Italian, the conjugated form of fare and the infinitive verb stick together like glue. You cannot separate them with the person doing the action like we do in English. The structure must always be a form of fare immediately followed by the infinitive.

If we replaced all'imbianchino with a pronoun to say "We had him paint the house", which pronoun would we use?

Because the painter acts as an indirect object in this specific structure, you need the indirect object pronoun gli (to him). You put it before the conjugated verb, so the sentence would become Gli abbiamo fatto dipingere la casa.

Why doesn't fatto change its ending to match the feminine noun la casa?