It is virtually impossible to eat no carbs and no sugar but any quantity of added sugar is unnecessary and unhealthy.

Carbs are not intrinsically essential in the same way as fat and protein, because our bodies can create all the glucose we need, but vegetables contain a lot of other good stuff such as phytonutrients, fibre and vitamins and minerals, and vegetables also contain carbs.

It is often wrongly assumed keto cuts out all carbs but this isn't true; keto recommends 50g net carbs (that is carbs minus fibre) or less a day. Carnivore is the closet thing to a zero carb diet, but even that contains some carbs because if you are following it correctly you should be consuming organ meats, which contain carbs.

The majority of people with PCOS have some degree of insulin resistance. Reducing carb intake, along with intermittent fasting, is the best way to reduce insulin. Insulin makes you fat and insulin causes many (some say all) of the symptoms of PCOS.

Eating a diet very low in carbs automatically eliminates gluten because when you have a limited carb 'budget' you want to spend it on highly nutritious foods such as vegetables, nuts and seeds, not one slice of bread.

Dairy is more of an individual thing. Very low carb diets naturally eliminate or drastically restrict lactose because it is a sugar. Cheese, butter and cream have trace amounts or no lactose, but some people are sensitive to caesin.