"What is your favorite thing to teach?"
My first question for any teacher I'm considering working with is What is your favorite thing to teach? This question gives the teacher the floor to talk about what he teaches best. It could be that he is the best at teaching jazz guitar, or loses herself in teaching lead rock guitar. If you want to learn lead rock guitar, then you've found your teacher.
Asking a teacher what his or her favorite thing to teach is also helps you understand what the teacher is prepared and organized to teach you in the first place. This includes what habits and processes she can impart to you, and what her curriculum is structured liked. If you ask this teacher to teach you something other than what she says is their favorite thing to teach, your experience might be limited.
What you are looking for is this: A teacher who loves to teach what you want to learn. If, for example, a teacher just absolutely loves teaching classical piano and all you wanted to learn was classical piano, you're all set! You're going to have a ton of fun! If a teacher loved teaching the blues and all you wanted to learn was classical piano, then it's best you find a different teacher.
"What is non-negotiable for me to learn if I take lessons with you?"
The question, What is non-negotiable for me to learn if I take lessons with you, is another important question to ask because it will help you isolate how the teacher will teach you and what requirements will be on your shoulders if you take lessons from him.
For example, some guitar teachers require that you learn how to read guitar music notation as well as guitar music theory, regardless of whether or not your path dictates learning them. In this case, it's important for you to understand what you're getting into beforehand, lest you make the mistake of working with a teacher who might not understand what you want to learn at all.
"What are you flexible about me learning or not learning?"
The question What are you flexible about me learning or not learning is a gauge of a teacher's flexibility in general.
Guitar teachers vary in their flexibility in what they are willing to teach you. Some guitar teachers feel that they must absolutely show you how to learn guitar scales. Wiser guitar teachers might want to show you guitar scales, but they wait for you ask before they jump into it. They want to make sure that you willing to learn, first. My recommendation is to find a teacher with a ton of stuff to show you, an encyclopedia of guitar knowledge, but one who is willing to wait for you to ask.
When you ask if a teacher is flexible about learning something or not learning something, you'll also get a measure for what this teacher is like as a human being. I'm sure you've met people in your life who you'd consider totally spoiled and without scruples when it comes to getting what they want. Flexibility is an indicator of a wise and compassionate teacher, one who gently understands the power and wisdom of collaborating with students.