Fat can only be stored when insulin is produced. Insulin is only produced when consuming carbs. The definition of insulin is, its a fat storage hormone.
The bolded part is very, very wrong. Insulin is produced in response to blood sugar elevation. You don't need carbs for this, the liver can kick out amazing amount of glucose on its own. Try drinking 3 scoops of pure whey mixed with just water on an empty stomach and measure your blood glucose every 10 minutes. You'll see what I mean.Gluconeogenisis is why we can handle low carb diets. Only the brain has to have glucose as its only source of energy, but its a pretty big deal if it can't get it. Fortunately the liver can step in and help.