Counting Veggies

Of course. Especially fruit since it tends to be more calorie dense than veggies (ex. 100 grams of banana is about 90 calories while 100 grams of carrots is just over 40).

Yes. Places like weight watchers purposefully give you too few calories with your daily points so you fill the remainder with fruits and vegetables. With standard calorie counting, they certainly do count, though some are more calorie-dense than others (bananas and avocados being on the more dense side, tomatoes and strawberries being less dense).