It doesn't say in the rules, so I just wanted to confirm - Are you allowed to bring any item you want into the levels? And/or for Claire levels, do times count with or without having gotten the ingredient first?

Oh you're right, I should specify in the rules, yes, you can bring anything into the levels, already have ingredients unlocked, and use secret exits. Only things that won't count are cannon exits since they don't count as a level completion according to the game itself.