I have a question regarding the Warforgeds Integrated Protection feature and the Arcane Armor feature: Which one takes priority when it comes to how long it takes to doff and don the armor?

Good question. If I were the DM, I would probably say Arcane Armor takes priority since in the beginning of the Warforged Integrated Protection it says that enhancements to the armor is allowed, and I would count Arcane Armor as an enhancement. 

An action, the racial states you're integrating the armor onto your person.

Arcane Armor states it's attaching to you, no integration necessary imo.