My Cat Is Breathing Heavily, She Has Gone Very Skinny Too, Being Fussy With Food. What Is Wrong?

Respiratory difficulty is an emergency.  Cats should not pant like dogs, open mouth breath, or use their abdominal muscles to aid in breathing.  If your cat is doing any of the above take it to a veterinarian immediately.  Increased respiration can be due to stress, pain, or primary respiratory issues.  
Not eating or drinking suggests nausea or discomfort.  It can be related primarily to the gastrointestinal tract or can be secondary.  It is always a problem when cats stop eating--they are very prone to developing a secondary disease called hepatic lipidosis or fatty liver disease.  Not drinking leads to dehydration which makes sick pets feel worse.
Your cat needs to be examined by a veterinarian to determine the cause of these signs and start appropriate treatment.

Hyperthyroidism is one of the possible problems with your cat.  Hyperthyroid cats do not usually become fussy with food--in fact they usually have an increased appetite that is accompanied by weight loss.  Heavy breathing is also not a common sign of hyperthyroidism.
Difficulty breathing is a cause for immediate concern.  Your cat needs to be examined by a veterinarian, have bloodwork run, and possible have chest x-rays taken.  Depending upon her age cancer is a concern from what you have described.

I agree with getvet. The symptoms sound like hyperthyroidism. Other symptoms include restlessness and frequent vomiting. My 16 year old cat has had hyperthyroidism for almost6 years.
The vet wanted use radioactive iodine treatment to destroy the thyroid and then I would need to give the cat thyroxine medication for the rest of its life. Not only would the treatment have been expensive, but try giving tablets to a cat!
Instead, I thought I would try a holistic vet and it was the best decision. The initial treatment required me to put some drops in the drinking water for a couple of months. The cat was stabilised for the last 5 years years and only recently have the symptoms flared up. He is now being treated again by the holistic vet. Much more gentle than the radical treatment proposed by the traditional vet.