Sounds like your dog is really uncomfortable. Skin conditions are far too common in dogs and unfortunately Lahsa Apso's are predisposed to
them. It also sounds like he’s been through the entire gauntlet of tests with no meaningful result, which is obviously frustrating for you. Skin conditions are by far the most frustrating for a vet to treat because there aren’t any definitive test and most treatments are based on a process of elimination.I’m going to give you some suggestions, which you should take to your veterinarian to discuss. Firstly: Have you been referred to a Veterinary Dermatologist? If not, I’d recommend it as they are skin specialists that deal with dogs like Gizmo all the time.
First question we need to answer: Why is Gizmo itchy? Possibilities include...
1. Allergies (Environmental/Food)
2. Secondary Infections: Bacteria, Yeast, Demodex Mites
3. Dietary Deficiency eg. Zinc
4. AutoImmune Disease : Pemphigus, Lupus
5. Endocrine Disease: Hypothyroidism

What you need to do with you vet is meticulously rule out all of these conditions.
Autoimmune Disease is usually diagnosed by skin biopsy or auto-antibody testing. You should definitely have the Thyroid levels checked as low thyroid levels can lead to skin conditions. Dietary Deficiency is uncommon but should be discussed with your vet. Skin mites like Demodex and Sarcoptes can be discovered with a skin scraping performed by your vet. Allergies are much more commonly the problem; they can be food related or environmental (Atopy), or a combination of both (very common). Intradermal skin testing (which you said was performed?) is the best way to diagnose environmental allergens Gizmo is sensitive to. There are Blood Tests for environmental allergies but they are far less reliable. Once the results of the skin test are known there are a few ways to respond:
a) Avoidance – exactly like it sounds, reduce exposure to allergens that you can like dust mites, wool, grass etc.
b) Hyposensitization Therapy: A series of injections from you vet that decrease the allergic response.
c) Immunosuppressant Therapy: Antihistamines, Corticosteroids, Cyclosporine A, Pentoxyphylline

Food Allergies: There is no reliable laboratory test for a food allergy. A food allergy must be diagnosed by a process of elimination. This involves starting a dietary exclusion trial for 10-12 weeks. A dietary exclusion trial involves feeding your dog an exclusive hypoallergenic diet that is designed to reduce food allergies. The special hypoallergenic diet only contains protein sources that your dog has not been exposed to previously. I’d STRONGLY recommend you use IAM’S RESPONSE KANGAROO AND OATMEAL FORMULA for the hypoallergenic diet, your vet can order it in if they don’t regularly carry it.

It takes a long time to become allergic to proteins in food so your dog should not be allergic to the new diet. By removing the proteins being fed in the previous diet we can remove the cause of the allergic reaction. If your dog’s symptoms improve while being fed the hypoallergenic diet it provides strong evidence that the previous diet being fed played a role in causing the allergic symptoms your dog was experiencing. It can take a length of time for all other food products to get out of your dog’s system so an exclusion diet must be fed for at least 10 weeks.The most important aspect of a dietary exclusion trial is that your dog be fed the recommended hypoallergenic diet EXCLUSIVELY. During the trial the only food that can be given is the recommended
hypoallergenic diet and water. You absolutely MUST NOT GIVE ANY table scraps, vitamins supplements, treats, rawhides, Dental Bones, etc. for
the duration of the trial. Even a small amount of non-recommended food can interfere with the results of the trial.

Dietary Challenge:
After the trial is completed a dietary challenge is performed. This involves reintroducing certain foods into the diet to see if they cause the allergic symptoms to return in your dog. Once specific allergen/allergens are identified it is much easier to create a balanced diet for your pet that does not contain the ingredients that they are allergic to. When a dietary challenge is performed a whole protein source is used. Good examples of protein sources to use initially are: Cooked Chicken, Beef, Pork, Lamb, Corn, Wheat (plain bread), and Cheese. Do not add dog treats or other food sources with multiple ingredients at the beginning of a dietary challenge as it is impossible to determine which ingredient is causing a reaction should one occur.

An example of a Diet trial with a dietary challenge:
After a strict 8 week hypoallergenic diet trial, one protein source is gradually added every 2 weeks to the hypoallergenic diet, twice daily. The new protein source is added exclusively for 14 days in a row and the dog is monitored for any adverse response such as Itching, Inflamed Skin/Ears, and Chewing at paws etc. If the protein source that was added to the diet for 2 weeks doesn’t cause a reaction it is assumed that the dog isn’t allergic to this protein source. We then challenge the dog’s diet again with a different protein source for 2 weeks. Dietary challenges are continued this way until the dog’s
diet is expanded sufficiently to have a balanced diet or until we recognize which foods the dog is allergic to. Now that we’ve been over things in painstaking detail: Here’s what I’d recommend. You vet feels that the cause of the problem is allergies (because the previously listed conditions have been eliminated).
Start on Prednisone on a tapering dose for 4-5 weeks of therapy. At the same time start treating for secondary infections: Your vet should be performing skin cytology on multiple areas including: Ears, between toes and other areas of dermatitis. This cytology is crucial to understanding what secondary infections are present. If you don’t treat for a yeast or bacterial infection the itching won’t stop. Appropriate antibiotic therapy or antifungal therapy needs to be for a least 4 weeks. At the same time as well, start the dietary exclusion trial (10-12 weeks). With this type of treatment protocol you are addressing all the possible causes of irritation at the same time, hopefully giving some relief. As time passes each therapy is tapered off and the reoccurrence of symptoms can give a strong clue as to where the problem lies.

Example: If Gizmo becomes itchy around week 8, it is likely an environmental allergy as you’ve discontinued Prednisone but are still on a hypoallergenic diet. I really hope you get some help for him, as you can probably tell from the length of my response. Like I said skin conditions are the hardest to treat. Let me know if I can help you further.

All the Best!!!

I'm really sorry to hear that your dog has had such serious health trouble at such a young age. Let me start by first saying that I am doubtful that your dog's limping has anything to do with the bladder stones. However, the bladder stones are certainly causing discomfort and that needs to be addressed.
Let's first address the bladder Stones:
I'd definitely recommend surgery if you can afford it. The stones may or may not be Calcium Oxalate, it's hard to tell with certainty on an xray, if they are Calcium Oxalate they won't dissolve with the diet, it's totally useless. If they happen to be a different type of stone like Struvite, they may disolve slowly... but in the mean time your dog is very uncomfortable, is predisposed to infection, has the risk of having a life threatening urinary obsturction, and is having further ongoing damage to the bladder. Have the surgery performed (cystotomy) and make sure the stones are sent away for analysis to a lab (you can't tell by just looking). This will get rid of the immediate problem, and identifying the stones will help you take the appropriate preventitive measures. After surgery definitely follow the diet your vet recommends as this plays a major role in preventing reoccurence. Medi-Cal/Waltham's Urinary SO is a good food for many urinary conditions, as well it's important to encourage water intake as this helps keep the urine dilute and flushes the bladder.
Secondly we'll address the limping:
The limping could very well be caused by Lyme's Disease as the bacteria can colonize the lining of the joints (synovial membrane) and they can be difficult to treat. Usually, there is an improvement in the limping with the initial antibiotic therapy (in your case especiall). Considering that you were also given pain medication it would be surprising if there was no change in the limp at all. Was there improvement??? If there was no improvement, you need to have your vet completely rule out other possible orthopedic causes of lameness including : Cranial Cruciate Ligament Rupture, and Patellar Luxation. These can be easily ruled out by examining the affected leg under anesthetic during the bladder surgery.
Has the vet figured out which joint it affected? You should at least be able to figure out that much by watching your dog walk, feeling the joints, looking for swelling etc. If your vet knows what joint is affected (eg. knee joint) and they aren't finding any problems with the ligaments, disc. obtaining a sample of joint fluid from that joint while he's under anesthetic. That sample can be analyzed to determine if lyme disease is the likely cause of discomfort, as it can become a chronic condition in some cases. The duration of therapy for Lyme's Disease is at least 4 weeks, you may need to go on a second course of antibiotics to clear the infection if orthopedic conditions have been ruled out as potential causes of the lameness. This is a start, hopefully helpful. I anticipate this will be an ongoing problem for the next while and I strongly encourage you to let me know if there is anything I can do to further assist you. Best of luck.

The first step would be to isolate where the smell is coming from as it rarely comes from the entire dog, rather it comes from a specific part of the dog. To be able to tell you need to actually get in close to you dog and have a smell. Start with the mouth: Look at the teeth, is there a lot of tartar and gingivitis? Is your dogs breath the same smell you've been smelling. Severe dental disease in older dogs can smell horrible. If the teeth are bad you can try using a water additive available at your vet's or at local pet store containing Chlorohexadine, also Dental bones and teeth brushing with appropriate dog toothpaste can help.

Not the mouth, next have a good smell of the ears and right behind the ears, do they smell musty and yeast like. Ear infections, especially with yeast can clear an entire room. If they smell bad I'd visit your vet again to see if you can get some medicated ear cleaner.

Smell your dogs coat, is this the cause of the smell? What does your dogs coat look like? It should be smooth and silky, not greasy with oily flaky skin. Oily skin and coat with lots of dandruff could be indicative of systemic illness like allergies or more serious illness like Cushings Disease, Diabetes, or Hypothyroidism. All of these conditions need to be assessed by your veterinarian. Definitely get back to me as to where you think the smell is coming from as this will help me help you further.

I know you said you thought it was coming through the skin, I just want you to check again to be sure. If it truly is through the skin, Allergies (Food and Environmental), Cushings Disease, and Hypothyroidism would be high on my list of problems.
Is your dog itchy at all????
When was the blood work/ Urine done? Do you know exactly which tests were run? It's seems strange that they were normal, with such sudden changes in drinking and urinating. Increased Drinking and Urinating is primarly cause by the following diseases: Kidney Failure, Diabetes, Cushings Disease, Diabetes Insipidus, and Psychogenic Polydypsia... has your vet ruled out all of these conditions?
Get back to me to help me help you further.