As you state, a ketogenic diet as not so much a quick weight-loss diet, but rather a permanent alteration in your eating habits. Its primary effect is the reversal of metabolic disease, and fat loss (if you have excess stored fat to shed) is a pleasant side-effect of the metabolic healing.
A well-formulated ketogenic diet is a low-glucose, low-insulin diet, because of the restriction of carbohydrate intake. Calorie-counting is not actually necessarily and can, in fact, cause your metabolism to slow down, if you restrict calories too much. The body adapts to the food we give it, and within certain very broad limits can adjust our metabolism up or down to compensate for our caloric intake. It is certainly true that the hormonal response to the foods we eat is far more significant than our actual caloric intake, but the metabolic response to a caloric deficit is likely to be to hang on to fat reserves, not shed them. Good Calories, Bad Calories, by Gary Taubes, and The Big Fat Surprise, by Nina Teicholz, go into the science of this in exhaustive detail, with numerous references to the primary literature.
Increasing exercise is not necessary and can, in fact, be stressful to the body during the initial couple of months after the shift to this way of eating. Give yourself time to adapt before returning to your exercise schedule. Exercise has been shown to be very unhelpful in terms of losing fat; it has many other benefits, however.
The point of not eating carbohydrate is that there is a threshold above which insulin gets stimulated and all the benefits of a ketogenic diet are put in abeyance at best, and at worst are reversed. Your carbohydrate tolerance may well be greater than the 20 g/day limit we advise, but you are better off, the lower you can keep your intake. Above your threshold, insulin rises to a level that is not healthful. Moreover, if your goal is fat loss, insulin is one of the most important of the hormones that cause the body to store and retain fat, so it is in general a very good idea to keep insulin as low as possible for as much of the day as possible. (We need a certain minimum in order to remain alive, but too much above that minimum is damaging to the body in many different ways.)
A day of higher carb intake here and there is probably not something to worry about, but carbohydrates can be addictive, and what we call “carb creep” happens all too easily. The harder you work to keep your carb intake as low as possible, the better off you are likely to be. Zero-sugar drinks can be helpful, if you need a crutch, but many people find themselves better off avoiding sweet tastes altogether (much as a sober alcoholic generally finds it advisable to stay away from non-alcoholic beer). There are also potential issues with most of the non-sugar sweeteners, not least being that they all appear to spike some people’s insulin. Whether a given sweetener will affect you is something that you can figure out only by trial and error.
As for how much to eat, eat to satisfy your hunger. Some days that will be a lot of food, other days, not so much. Don’t worry about it. In the absence of insulin, your appetite-control hormones (ghrelin and leptin are the major ones, but there are several others) will work better as a guide to how much to eat. One study I read showed that over any seven- or eight-day period, ad libitum food intake correlated astonishingly closely with energy expenditure. In general, most people can trust their appetite to lead them to eat enough food, but not too much. And if you have excess stored fat to shed, you will be satisfied with an amount of food that will allow both the fat you eat and some of that excess fat to be metabolised. I personally lost somewhere between 27 and 36 kg by not counting calories.
The last 15 kilos will take longer to come off than the first 150, so give the process plenty of time. Keep calm and keto on (KCKO), as we say! 
