Hi Pavel, yes we do use this expression in English, although where I come from we say it without the French accent 'on route'. It means that you are on your way to somewhere. For example, if you receive a phone call on your mobile from your parents asking if you are on your way home, you could reply 'Yes, I'm en route', which means that you are on your way home. 'By the way' is something completely different.

Yes, it's very common. It's pronounced 'on route' , and it means that you are part-way through your journey to a particular place. 'His car broke down when he was en route for Birmingham.' It can be replaced with 'on his way', 'on your way', 'on their way' etc. The expression 'by the way' has nothing to do with travelling. We use it as a discourse marker in speech when you are introducing a new topic which isn't necessarily related to the previous one.

Yes. Where I live we pronounce it EXACTLY like the words "on root."