Do babies brains continue to develop after they're born?

Yes. Technically, they never stop developing; every time you learn something new, you develop new connections in the brain, be you a month old, or 90 years. The rapid maturation goes on well into the teen years.
Yes. The brain develops a great deal after birth. There are many important connections and pathways formed in the first 5 years. Our judgment centers are not fully developed until we are 25, and as mentioned before, we all continue to learn and grow and develop all through life. Reading to our children helps develop language pathways which are so vital in our development.
Yes. Some classify this into 4 basic stages: approx 0-7 an age of fantacy and innocence(the tooth fairy works); ~8-14 an age of concrete thinking(tooth fairy doesn't work);~15-23 the i understand it all, everybody needs to listen to me stage ; ~24- stage of realistic thinking. The stages are thought to represent realizations caused by nerves making cross connections in the brain.
Yes. Definitely! that's why nutrition is so very important in children. The development of your child's brain is not only a function of his nutritional status, but also a reflection of the stimulation you have provided to him. Be sure to read to your baby every day, talk to him, play music, smile, give him hugs. All of these factors will benefit his brain development over the years.
Yes. Absolutely. Brains grow and develop through childhood, and into early adulthood. Not only are new things and new skills learned, but more-sophisticated ways of thinking and interacting with the world continue to develop. It's an amazing process.