Although I’ve never actually heard anyone say this on their own in France, my English language students have confirmed that it’s an actual expression that everybody knows.
If you have your “head on the handlebars”, you are focussed single-mindedly on a goal or overwhelmed by an activity. It can also mean you are busy, I think. Although this turn of phrase definitely comes from cycling, I don’t think it’s use for cycling – at least I’ve never heard it in race commentary. It’s figurative.
I think most of you will understand this one.

Heh… I’ve actually done it a couple of times. On the trainer, of course. A little too dangerous to try on the road.
I have never heard the expression.