There are many things to consider when answering each of your questions. I will list each question you have asked and then go from there.
1. What is the prognosis for anesthesia of a 14 year old dog for MRI and surgery?
This is influenced on the rest of Katy's health status. Age is not a disease itself, although it is a factor that animals that are older often have other underlying conditions, such as kidney, liver, or heart disease. This influences the type of anesthesia that is used. Any time there is anesthesia, there is risk involved, no matter the age of the animal or person. That being said, anesthesia in animals is much safer than it has been in the past. I just anesthestized a 20 year old cat for a very advanced and involved dental procedure on Thursday - he did fine with the proper mix of anesthesia, monitoring, and fluids. Some of the prognosis also involves the disease process that is going on in her neck. Sometimes, the MRI shows that the area is not able to have surgery after all. Anesthesia itself is likely to not be the problem.
2. Will the surgery remove Katy's pain?
Once again, this depends on the problem with her neck. Usually, after the post surgical pain is gone, pain is significantly less. Thankfully, we do have good pain medications to help them. As you mentioned, there is no guarantee that the pain will be gone and that she will have range of motion.
The questions that I usually ask my clients include:
1. If you do not do the surgery and she lives for another 3 years, will you feel guilty that you did not do the surgery?
2. If you do the MRI and surgery and Katy dies in 6 months (or even during the surgery), will you feel guilty that you did the surgery or angry that you "wasted the money"?
3. If you do the MRI and surgery, will you be able to feed your family and keep your home? or If you do it, will you lose everything?
The reason I ask these questions is not to make you feel bad, but to make you take stock of the situation. If you will lose everything, then I do not think that surgery is the right thing to do. If it means you can't go on a fancy vacation for 2 years, but you can still feed your family and pay your bills, then you are in more of a grey zone. But, honestly thinking about your reactions to each question will help you make the decision.
Most of the time, in my experience, the surgeon will go straight from the MRI to surgery, no waking up in between. This makes it safer as the most dangerous parts of anesthesia is the induction time and when they are waking up. So asking about this is important. Sometimes, they do need the time by waking the patient up to make more definitive plans on how to pursue surgery itself. You can also ask for an answer on the MRI before going to surgery. If there is a massive tumor that is causing the problem and removing it could cause her to be paralyzed, then waking her up without surgery or euthanizing her without waking her up may be the more humane thing to do. As you can see, there are many variables to take into account.