Any amount of cardio is good However to optimize fat burning it is best done for at least 20 minutes and first thing in the morning on an empty stomach. Following a low carb WOE we have an advantage over most in this respect. Your body will first use carbs for energy when doing aerobic exercise followed by fat, the greater your store or carbs, the less fat will be burned. By working out first thing your carbs are at their lowest as you've been fasting (sleeping) all night. It is estimated you can increase fat burning by up to 300% this way.

As you stated, all muscles need to rest. A good rule to follow is 2 days rest between workouts. I'm sure the experts here will be able to provide some great exercises to follow and some ideas on sets and reps. But remember that in order to see the results of all your hard work, your body will have to get rid of the fat. Low Carbers have an advantage here too, our diets are rich in protein which our body uses to build muscle. I've noticed a marked improvement in muscle tone since I started resistance training (at a much faster rate than before). The recovery period seems to have speeded up as well.

Best of luck to you!
Nat