There are vegetables that are high in carbs and others low in carbs. The keto diet recommends sticking to the ones low on carbs but encourages you to eat a lot of them. Best vegetables are all green ones to make it easy. And vegetables that grow above the ground (e.g. lettuce) are always better than the ones that grow below the ground (e.g. potatoes)
It will take a while to trust and adjust to your new appetite. I personally found when I was low fat and on Weight Watchers, I was ravenous all day long, so to be told eat what you like was difficult to understand. By eating more healthy fats especially, will keep you fuller for longer. And depending on how long you have been dieting for, and how insulin resistant you are, will depend on how long it takes for you to adjust appetite and weight. The odd person gains weight when they start because they don’t listen to their body. Remember to eat until 80% full (it takes a while for your stomach to register you have eaten your meal) and only eat when hungry. Don’t snack if you don’t have to. Are you really hungry or thirsty (similar feelings).
I personally don’t count anymore as I want this to be as easy and sustainable as possible. I have had years of counting calories and points, and this is incredibly liberating. I just don’t eat any sugars, grains or high carb foods any more so I am incredibly low carb all the time. When I was starting out I counted, just to see where my carbs were coming from and it was an eye opener. And yes you are correct, to go into ketosis anywhere between 20-50g carbs/day. Find out what works for you.
The reason why low-carb plans so often fail most of us is that they're not sustainable for the long term. They often fail to provide a "fallback" plan for what to eat when low-carb foods aren't readily available. Birthdays, holidays, work functions … there's likely at least one scenario in which you'll find yourself eating high-carb foods that don't necessarily "fit" into your plan.
Now if you're thinking you'll just handle the problem by brushing and flossing a little more often, guess again. Since the breath odor is coming from metabolic changes and not necessarily a dental-related condition, traditional breath products are not likely to provide long-lasting relief. On the other hand drinking more water intake can do the trick.