My neighbor stole our doormat so I'm going to take the one they have in the back of the house and throw it out.
Thats rude. Why would you feel the need to throw out their mat?
Retaliation. I want them to know how we felt.
Being upset that they stole your doormat is fine but you know that two wrongs don't make a right. You shouldn't be throwing things away that don't belong to you.  Instead either confront them about it or just move on with your life.
I'm going to have my brother frame them too.
Do you have proof they stole your mat? Otherwise you're just punishing them without any actual evidence. Doing so will just upset your neighbor and they may retaliate back.