Everything goes in the bike box. When I’ve packed my bike, there’s a nice little cavity in between bits of the frame where the helmet fits perfectly. 🙂
In theory, all the budget airlines have a “you should only pack the bike in the box, nothing else” but I’ve flown dozens of times with everything shoved in there – clothing, a few tools, helmet, toiletries, cycling shoes… No-one has ever said anything about it, it’s always been under 32kg and everything has always arrived safe and sound.
Only time it was ever opened was TSA as I transited through USA but even then they managed to leave it all undisturbed and close the case up properly.