With the All Story Mission runs, are they counted solely as a single play-through? Or as being completed twice over to collect all Minikits, Character Tokens, Red Bricks, achieve True Hero, and save Adam West.

All Story Missions is just a single play through of the main story. It used to be called any% until a major skip was found where you can unlock every level after finishing the third level.

In 100% you will have to collect all minikits, character tokens, red bricks, achieve true hero, and save Adam West as well as do anything else to get the percentage counter to 100%.