"I do it easily."
Translation:אני עושה את זה בקלות.
That's a good question with no easy answer. אני עושה זאת is a bit outdated, but sometimes still used in writing as a fixed form. It looks like this, and rarely אני עושה את זאת (while you would never see אני עושה זה), I think it's because of the nuance of the meaning of זאת in older Hebrew.
I agree, except I don't think עשה זאת is a fixed form. זאת can be used instead of את זה as the direct object in a sentence, and while it's more common with the verb עשה it can be used with others. "כאשר שמע זאת יעקב, הוא החליט לצאת מיד לדרך" or "הוא סוף סוף אמר זאת" are okay as far as I know.