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hey, awesome work with the speedrun!
Each of the stats (Prof, Competence, Fun) don't affect combat in any way. Each stat is used for stat checks in Wrong Turns for different obstacles (unopened giftboxes for example) and yield different results (rewards like food, ammo, etc). Fear Level functions the same, but chiefly with opened giftboxes (i believe above a certain level, you will always fight something)
You're also right with Competence letting you ignore broom dmg too as well as optional areas like the Playground and Freezer.
anyway, hope that all helps and good luck with the speedrunning!