You asked

Does the eye examination hurt?

The eye examination doesn’t hurt at all and all the tests are child friendly. Your child doesn’t have to be able to read in order to get their eyes tested.
The ophthalmologist or optometrist will first gather information about family history and ask you whether your child has any problems playing games, or seeing small objects. He will then check for nearsightedness, farsightedness, astigmatism, amblyopia, proper eye alignment, and how the eye reacts to light and dark. 

More questions

Children's eyes are sensitive so it is vital to protect them from the sun's harmful rays.
Your young child can have their eyes tested even if they can't read
Regular eye exams are essential in order to monitor your child's eyesight.
There are some signs to watch out for if you think your child may have a vision problem. 
Should your child get any kind of chemical in her eye you should treat it as an emergency and bring her to the hospital immediately.
You should take your child to their G.P. or to the hospital if you notice any of these symptoms. 
You should seek immediate medical attention if an object seems to be imbedded in your child's eye. 
If your child appears to have something in his eye you should seek immediate medical attention if it is imbedded in the eye. 
As your baby learns to crawl and begins to explore her surroundings you will need to child proof your house to prevent eye injuries.
Babies and young children are always getting up to all sorts of mischief and eye injuries are quite common. It is important to always consult with your G.P. should any eye injuries occur.