At our school, they let you request a specific teacher. But people from other schools seem shocked when I say that.

It depends on the school. You can ask, but that doesn't necessarily guarantee anything. Another idea would be to discuss her skills with whatever teacher is assigned to her, so that they can provide support for her individual needs.

I want to know how kids get sent to one age group-based class or another. Some schools have 3 year-old classes, some 4 year-olds, some mixed. How does that assignment work? if I have a girl with great fine motor skills at age 3 before entering school, ready to write - can I ask that she go into the mixed three and four year-old class?