You can say "I'm on my way" to explain that you are travelling and it is obvious from the context where you are travelling to. If it's not obvious, add an adverb or expression place e.g. "I'm on my way home /. to you / there / to your house / back" In your first example, you need to add "home" because otherwise, I would not be sure where you were on your way to. "When I get home" refers to the time when you arrive home. You don't need to add the word "back" but sometimes people do.

Yes, you can say "I am on my way" to mean that you are in the process of going to somewhere. It is a common thing to say. And you can say "get home" or "get back home" in that situation. I think that "get back home" has a bit more emphasis, say if you are on a long trip and have been away for awhile, you could say "I can't wait to get back home". If you are just coming from your work or a friend's house "get home" would seem more natural.