Don't think INZ are going to assume ANYTHING. They don't know you. To be blunt, your daughter is evidence that you spent time together ONCE, and all the rest of your time living together remains to be proved.
INZ are very likely to be looking hard at the circumstances that caused your Australian visa to be cancelled, yes. Immigration services around the world are in contact with one another. To the official mind, someone who broke the law in one country is a person who may be likely to cause trouble in their country too. You may get questioned about it.