Indeed, this is a very flexible language. You can have the subject an the end of the sentence, the object in the beginning, you can start or finish a sentence with a verb. And thanks to its grammar's complexity, everyone will understand what you are trying to tell them ;)
This is not quite the same. “I didn't do it” might be the tense we use more often in English, but if we translate that into Greek it would be: “Δεν το έκανα εγώ.” The difference in meaning between the two sentences are minor but significant given the lesson.
No, κάνει comes after έχω so cannot end in -ω. As in English, the auxiliary verb shows the person (e.g. I have done something, he has done something) but the second verb keeps the same form. Review a conjugation table and you'll see which tenses use κάνει.