Should I have a doctor examine my child's eyes?

Yes. Your pediatrician should check your baby's eyes at every visit. If he/she finds anything of concern, you may be referred to a pediatric ophthalmologist. Most pediatricians will also start screening for visual problems at age 3 or 4 in their office. Again, if there are any signs of problems, you will be referred to a specialist for a more complete exam.
Yes. Dr. Lindor's answer is right on target. I have nothing to add to her excellent answer.
Yes. Your pediatrician should check your baby's eyes at each well visit. Once your child is old enough, formal visual acuity screening should be done at each well visit too.