yes it is, just make sure you put a cotton textile on his back to prevent disease and make sure to give him water at most times to prevent dehydration. to make sure, adjust the conditioning of your room at night
No, it isn't normal, but there could be a few reasons. Is the temperature in the house to warm, or the room itself? What do you have him/her dressed in, or also, they might just have a temperature.
First, make sure it's not too warm in the room for your baby. Infants' bodies can't regulate their temperature as well as older children and adults. Dress your baby in cotton sleepwear at night, and infants should not have blankets as they pose a suffocation hazard.