Karate-ka - Karate student Sensei means "teacher" Sempai means "Senior" (as in senior karate student)
Where ever his teacher is located. And they all have a teacher. If their master has passed away, they train on their own in their own dojo.
That will depend on the school and style. Each as specified the skills required. You will have to consult your school's instruction manual, or talk with your teacher.
The teacher in the karate kid movie
Yes! Actually, my karate teacher taught him some moves--no joke!
Karate has many forms of practice, freestyle karate is only used in competitions and exhibitions, is a type of karate mixed with gymnastics, there is no default for freestyle kata karate. It is a kata devised by you or your teacher.
Yes there is such thing as a karate book but if you want a specialty you have to be training. You to ask your teacher to give you one.