Quite normal. While he is sleeping and cuddled up and warm his nose will feel warm. When he gets up and starts playing again his nose will feel cold. The fact that he is eating and playing and then having sleeps shows he is a perfectly normal little puss.
Here is a link about teething. Yes, he will normally have all his adult teeth by about 8 months. You may find some of his tiny milk teeth from time to time when they fall out, but it's more likely he will swallow them.
Have fun with your little cat.