First of all, welcome to the forums!
Summary: if you are truly keeping your carb intake low, trust your body. But carbs can sneak into our diet, so if you’d llke help ferreting out hidden carbs, we can help.
Now, on to the biology: The point of a ketogenic diet is to lower insulin levels to the point where it becomes possible to mobilise our excess fat to be metabolised. This requires minimising carbohydrate intake, because carbohydrates are nothing more than glucose molecules bonded together in various ways, so that eating them significantly raises our serum glucose. And that excess of glucose needs to be driven from the blood (or it will cause significant damage, and can even be fatal), which is one of the many jobs of insulin.
Unfortunately, one of the things elevated insulin causes to happen with that extra glucose is to make the adipocytes (fat cells) store it as fat, and not let it out until insulin drops. And on a high-carb diet, it never does drop. But, while ketones are valuable both as fuel for the body and as signaling hormones with many beneficial effects throughout the body, they are also valuable as an indication that our insulin level is sufficiently low, since elevated insulin inhibits ketogenesis. The ketone bodies are amazing molecules.
As you have guessed, in the absence of dietary glucose (carbohydrate), the liver does use a certain amount of the amino acids (protein) we eat to make the small amount of glucose that the body actually requires (about 1 U.S. teaspoon circulating at any given time). But this process is highly regulated, and is more determined by demand than by the supply of protein. However, as you have also guessed, if we do not eat enough fat to produce the needed ketones from, the liver makes extra glucose to feed the brain.
The brain loves ketones, and will use them to the extent that they are available, so it makes sense to avoid damaging our brains with too much glucose and insulin by making sure that ketones are available at all times (this means eating fat, of course, since there is a limit to how much fat can be consumed from our fat stores in a day).
Also, since fatty acids are too large to pass the blood-brain barrier, the brain also uses ketones to make the fatty acids it needs (the brain is made largely of fat, and a surprising percentage of that is cholesterol). Cholesterol, in particular, is required for the proper transmission of nerve impulses, so a high-carb diet not only damages brain cells directly, it also interferes with nerve impulses. This is why a number of researchers believe that a ketogenic diet can be of great help to people with mental health problems and those with motor-neuron diseases.
'So keep your carb intake strictly limited, eat enough protein for your body’s structural needs, and consume enough fat to provide energy for the body’s activities. Trust that your body knows what it’s doing. And, as mentioned above, if you suspect that carbohydrate might be sneaking into your diet, we can help you figure it out.