1. You can train every day, but you need to make sure that most of your running is very easy. Easy in this case meaning that you aren't feeling it too much the next day.
2. If your legs are hurting badly, more than just feeling a little tired, then you are running too fast or maybe too far. Find a speed that you can run the distance without hurting too much. As you get fitter this will naturally increase.
3. Lower carb diets are fine as long as you're not doing high intensity workouts, eat plenty of veg and generally healthily and you'll be good.
4. Train slower, of the miles you do each week, 70-80% should be easy. Probably 9-10 min miles for you. Then for one or two sessions a week up your speed, but when you're a beginner, most of the fitness gains come from running plenty of slow easy mileage.