The Innocent Farm, Curry, and the Alchemy shop all progress while the player is off fighting. Does going into the Character world contribute to this progress or is that progress only affected by actual fights.
Yes it does.
Each trip into the character world would equal 1 day, each stage is 1 day, and the only exception to that is the Item World is 1 day per floor you complete/skip (not including floors skipped by diver items). The assembly is the only thing that depends on whether you fight or not. If you try to pass a bill and fight them, it will count as 1 day, otherwise it doesn't.
So think of it as any fight/stage (including Chara world) is 1 day. This is noticed more when you have some curry sitting for up to 100 days (max), because stages like Character World will take up 1 day.