Hi Colleen -
I'm a professional longarmer, and while each of us runs our businesses differently, I'll tell you how I do things.
When I'm working with a new client for the first time, I will ask them a lot of questions, especially if they have not worked with a longarmer before and don't know what to ask. And there are certain things that we can or cannot do that you may not understand. (Like why I need the backing bigger all the way around. Or why certain battings are not suitable for frame quilting.) I'm happy to answer those questions for my clients and show them my studio and the supplies I use.
I have a page on my website that might answer some of your questions, or at least give you some ideas of the things to ask:
http://www.andicraftsquilting.com/longarm-quilting.htm
To answer some of your questions, Yes, I would ask to see samples of their work. When I show those to my clients, I ask which they like or don't like and why. It helps me suggest patterns that would look good on their quilt. Do you like more open patterns or denser patterns? Is the quilt for a man, woman, child? (I'm probably not going to put hearts and flowers on a quilt for a 16 year old boy LOL!). Do you like the thread to show or to blend in? All of these will help me give you a finished product that we can both be happy with!
A good longarmer will not be critical of your work. She *will* tell you if there are 'issues' that will prevent her from giving you back a quality finished product. For example, if you have 4 extra inches of fabric in your borders, it will be hard to quilt without putting pleats and puckers in. I will suggest ways to fix the fullness, and may offer you my borders handout (see here:
http://www.andicraftsquilting.com/learn.htm )
so you will not have this fullness in future quilts. (I also teach quilting classes so this is a perk for my clients - a mini class on borders or bindings or whatever.)
The studio should be clean, non smoking, and if pets are present (which I do NOT allow in my studio, even though I am a pet owner) you should be made aware of that in case you have allergies. Does the owner have business insurance in case something happens to your quilt while in her care?
I hope this helps you a little. If you have more questions, just ask!