Good evening! My name is***** and I am happy to help you with your questions! There is a very thin, superficial muscle called the Latissimus Dorsi that runs along the back of the dog. We actually purposely pinch this muscle in certain spinal injuries to see if it will twitch. When the skin twitches it is because you have stimulated the nerve causing the muscle to contract and it is a perfectly normal response. Depending on where the twitch stops we can determine which nerves are intact and therefore where the spinal injury is located. Contraction of the Latissimus Dorsi is the most likely the cause of the skin moving if it is happening when you touch his back or sides. As far as the corners of his mouth pulling back as if he is smiling this is a common response that we see when their back is scratched (along with licking the air and and scratching at the area with the back legs). It is basically a response indicating that the nerves are being stimulated and causing an itching sensation. I would consider these sorts of responses normal but they can be exaggerated if there is a skin issue that is causing the skin to be extra sensitive, itchy or irritated. If the skin itself looks normal (no redness, flaking, scabs), this is not cause for concern. Regarding, your description of his "legs almost buckling", there is no specific issue that would explain it but it could be related to the responses I discussed above. If it only happened once it is reasonable to monitor for recurrence.
As far as the mass, unfortunately it is impossible to know the type of mass without further diagnostics. Many times veterinarians can make assumptions based on the appearance. For example, soft tumors beneath the skin are often lipomas (benign fatty tumors) and certain tumors above the skin are often cysts (which are also benign). However there are more serious tumors such as Mast Cell Tumors which are cancerous and can look a lot like cysts or lipomas to the naked eye. If you are concerned, I would recommend insisting that your veterinarian run some test on the mass. He or she could start with a needle aspirate. It is a very simple procedure where a you stick a needle into the mass, express the contents onto a slide, stain it and look under the microscope. Many tumors can be easily diagnosed this way (including mast cell tumors and lipomas). If it is a cyst a lot of times you can drain out the fluid and the mass will completely disappear. In that case you know you have nothing to worry about :). There are certain tumors however that will not exfoliate well into the needle and in those cases you actually need to biopsy or remove the mass and submit for histopathology to get a definitive diagnosis.
Because you said the twitching and smiling happens when you touch around the mass and that it has grown, I am wondering if the mass may be invading the underlying Latissimus Dorsi muscle and stimulating the nerves within it causing it to twitch. Although, as I mentioned, you cannot diagnose the type of mass based on appearance, cancerous masses tend to grow quickly, become hard, and also frequently ulcerate (bleed or scab) or become infected. If you notice any of these signs you may want to be more aggressive about having your vet get a definitive answer for you. Also if it is bothering your puppy, you could have it removed regardless of whether or not it is benign.
I hoped I have helped answer your questions. If there is anything else I can do for you please let me know. A husky/lab is a very interesting mix! I'm sure he is a great dog!!!