Question 1. Argumentation is based on logic. While Persuasive writing is based on emotion.
When two people argue (not a logic thing to do) but let's just use it anyway... They are logic in their arguments. Yes, it may not be logical to you or I, but it's a logical argument
When writing a persuasive paper, you the writer are trying to pursuade I the reader of what it is you wrote about regardless of fiction or non fiction.
Question 2. Rhetoric
This is a process of debating induced with the logically thought process in a particular subject matter topic or discussion.
Question 3. A counterclaim is when a statement that responds to and stands to defend against another claim or accusations.
Good luck
I hope this helps