You would definitely want to limit sugar on keto. I use a small amount of coconut sugar here and in the recipe on my website, I note that it is optional or you can use another option. That said, keto looks different for everyone. So a very small amount of coconut sugar or maple syrup may fit into your approach. As for carrots, they are a whole, real food, they can certainly fit into a ketogenic approach. The amount would be dependent on the individual and their goals. If you are wanting to be CRAZY low carb for a short period, it is certainly a food that you may want to limit, because it is higher carb than other vegetables, for sure. Hope that helps.
Nope, it's most certainly not. I use Marukan Organic Rice Vinegar, contains ZERO sugar in the vinegar. No carbohydrates and no sugar. Not entirely sure what brand you use or have seen, but that is not a true statement across the board.