Does exercise affect the amount of carbs you can eat on the keto diet? ?

No. The body still has to process the carbs, whether you exercise or not. I'm not a fan of the keto diet, but your carb limit doesn't change due to exercise. 

Nope. Exercise and or calorie burn aren't relevant to being in or staying in ketosis.  They will impact your calorie deficit, of course, which is an essential component to weight loss.

Keto is a lifestyle, not just a weight management tool.  The "value" of keto is that you get your carb intake low enough so that your body starts burning ketones for fuel rather than glycogen.  This enables your body to burn stored fat more easily and also eliminates most of the health issues around high blood sugar and insulin.

The only way to truly know if you are in ketosis is to test your blood, but very few people get that detailed.  The general rule is that you should eat no more than 50 grams total of any type of carbohydrate and on 20 net carbs per day.  Net carbs are the total carbs minus fiber basically.