Hello, my name is***** and I have over 20 years of experience as a veterinarian. I understand your concern about Spooky's behavior. It's no fun to find stool where it shouldn't be.
Cats stops using their litter box for a few reasons:
1) They don't feel well and associate pain with their litter box so go elsewhere hoping not to experience discomfort.
2) Their litter box is dirty, hard to get to or in or out of, they don't like the litter in the box, or they don't have enough privacy in it or feel trapped in the location it is in.
3) Other cats (can be indoors or outdoor strays), dogs or people are scaring them when they are in the box or trying to get to the box.
4) Social stress and overcrowding. Cats in the wild do not live together. It is socially stressful for them to have to be confined together. That is likely why many cats stop "marking territory" once they are allowed to go outside. They are less "socially stressed" because they have more room and get more exercise.
First I recommend limiting access to the area that she is inappropriately using as a place to eliminate. The longer this goes on the more it becomes a habit. If this has been going on for a while it is likely that we need to find and treat an underlying problem that started all this as well as retrain her.
Ideally she should have a physical examination to make sure all is well. Make sure her anal glands are checked, and that she doesn't have parasites (check a stool sample) or spinal arthritis that makes it painful for her to go or maintain her position.
If she ever has blood or mucous in her stool, it is soft, hard or very large or small and difficult to pass she may have inflammatory bowel disease, constipation or megacolon which are all uncomfortable. We need to address any medical problems to have hope of retraining her successfully.
You also need to make sure that the areas that she has picked to go have been cleaned with an enzymatic cleaner, like Nature's Miracle. Although it may smell fine to you their noses are much better than ours and as long as the odor is there she will be attracted to the spot.
Make sure her box(es) is/are spotlessly clean, scoop stools daily, change litter completely weekly and clean the box itself. If the litter box is older than a year get a new one. Many cats don't like odor and an old litter box stinks to them, so i am glad that you bought a new one. It might be fine to pass urine there as that is quick but stool takes longer so they are more particular. I you should have 1 more box than the number of cats, 2 boxes if she's the only kitty. They should be in different locations and where she cannot be bothered and they are easy to get to. Some cats like to urinate and defecate in different areas and are very sensitive to being interrupted. I know you have done some of these things already, but I list them all to be complete.
Make sure she has privacy when she goes, yet also make sure the box is easy to get to and get in and out of. Many cats appreciate low sided boxes, especially as they age. Some cats also like a bigger box to pass stools in so they have plenty of room. The plastic very low sided storage containers that fit under the bed work very well.
If you are using scented litter I recommend plain clay litter or plain scoop litter. Most cats find scented litter objectionable. You might try different types of litter. Some cats have different preferences for urine and stools.
To help ease social stress you can try using Feliway sprays or diffusors. These are synthetic pheromones which mimic those produced to mark areas as safe and many cats find them soothing. You can also use pheromone calming collars as well. See this link for some examples: http://www.amazon.com/Sentry-Behavior-Pheromone-Collar-Inches/dp/B0026JAKWG
If these measures aren't enough you can try a homeopathic calming oral medication called Bach's Rescue Remedy. See this link for further information: http://www.bachflower.com/Pets.htm
And you could discuss oral medications with your veterinarian as well such as fluoxetine or amitriptyline as calming agents to decrease her stress.
Finally if she does better when allowed to go out you may wish to construct an outdoor cat pen so she can safely spend time outside. Here are some examples:
If you do all this and she is healthy and she is still not cooperating I would confine her to a large dog cage or small bathroom with her food
, water and litter box to retrain her. You can let her out only when you can supervise her behavior. If you catch her going to her "spot" use an air horn to scare her. She must have the negative consequences every time or this won't work. Do this for several weeks until she consistently uses her box. Then slowly give her more access to your home, a little more area each week if she continues to behave.
Best of luck with this girl and please let me know if you have any further questions.