I am concerned that Patches is getting very thin even with a good appetite, is vomiting hairballs frequently, and is literally pulling out her hair.
Her increased vomiting of furballs is probably related to her excessive grooming, but there may be additional reasons for more frequent vomiting. Increased grooming itself can be a stress or anxiety response, she may be grooming due to an underlying disease process or because she doesn't feel well. Some kitties with IBD groom themselves excessively in a response to abdominal pain.
When we see weight loss with an apparently normal or increased appetite we need to worry about disease processes that either don't allow her to digest and/or absorb the food she is taking (such as diabetes, liver disease, primary intestinal or pancreatic disease including cancers) or those that lead to a higher metabolic rate such as hyperthyroidism or cancer.
In an older cat the more common diseases we see include hyperthyroidism (a tumor of the thyroid gland that overproduces thyroid hormones) or if she were drinking more and urinating a tremendous amount, diabetes. Sometimes early on with diabetes we only see weight loss and irritability. Both diseases will cause increased hunger because they lead to calories being burned at a rapid rate or not being properly absorbed, thus cats tend to eat too quickly or overeat and vomit.
Both diabetes and hyperthyroidism are treatable or at least manageable.
Hyperthyroidism is controlled with daily oral medication (Tapazole, also called methimazole) or a very special diet called y/d, both of which must be done for the rest of her life or treatment with radioactive iodine to selectively kill tumor cells which is usually a one time therapy. There are pros and cons to each therapy which your veterinarian can discuss with you after getting a firm diagnosis.
Hyperthyroidism is not painful, but elevated thyroid hormones can make them feel edgy or nervous and thus cranky, may make them hypersensitive to touch, cause excessive grooming in an attempt to calm themselves, increase the heart rate and interfere with sleep, like drinking way too much coffee or energy drinks. That could explain her excessive grooming and her weight loss. Long term if left untreated it can lead to heart disease, hypertension, possible eye damage as well as kidney and liver damage. But if caught and treated many of these things can be reversed.
Diabetes is controlled with diet and insulin therapy.
Has she had any testing done to achieve a diagnosis?
Ideally she should have a complete blood count and biochemistry profile with T-4 checked as well as a urinalysis. These tests often give us the information that we need.
If those aren't diagnostic then it is time to move on to more specialized testing. I recommend checking vitamin B levels to look for signs of intestinal disease, a test for pancreatic insufficiency called a TLI, and an abdominal ultrasound to look for signs of changes in the walls of the intestines as well as evaluating her organs. In some cases we may need an endoscopy to collect biopsies of her intestines if they look abnormal on ultrasound, although many people understandably choose not to have their older kitty undergo that procedure.
In the meantime you might try feeding her a canned variety, higher calorie, prescription food called Hills a/d or Iams Maximum Calorie from your veterinarian and make sure she has access to plenty of fresh, clean water. Cats tend to vomit more when eating dry food, especially voracious eaters, as dry food expands in the stomach and that can cause them to vomit.
Please let me know if you have any further questions.