Learn Italian: Marco fa pianificare il matrimonio all'organizzatore.

This is the key rule for the fare + infinitive construction! Because the action pianificare (to plan) has a direct object (il matrimonio), the person actually doing the planning must be an indirect object introduced by the preposition a. Here, a combines with the article l' to make all'.

What if Marco was having the planner just "plan" in general, without mentioning the wedding? Would we still use all'?

No, and this is a crucial difference! If there is no direct object like il matrimonio in the sentence, the person doing the action becomes the direct object instead. You would simply say Marco fa pianificare l'organizzatore (Marco has the planner plan). You only use the preposition a when the direct object spot is already taken.

Could I change the word order and say Marco fa l'organizzatore pianificare il matrimonio like in English?

No, you must keep fare and the infinitive together. In Italian, fa pianificare acts like a single verb block. You cannot split it by putting the person in the middle like we do in English. The standard structure is: fare + infinitive + direct object + a + person.

Does fa pianificare mean he forces the planner, or just asks him?