1) Yes, the words "tutor" and "tuition" are related. They both come from the same root in Latin.
2) As for the meaning of the word "tuition" - that may vary depending on the part of the English-speaking world in which you are working. In some places, "tuition" is the act of teaching, or even the act of keeping a student at a boarding school. In other places, such as the USA, the meaning of "tuition" has morphed into the fee that is paid for instruction...in the sense of "The tuition of certain schools is astronomical, while at others it remains reasonable". In such places - where "tuition" means "fee" or "price" - the act of teaching is either "instruction" or "giving lessons". (These are interchangeable.)
3) So, if you were planning on being a "tutor" (that is, one who teaches OUTSIDE of regular school or class hours, to provide special assistance to one or more individuals), in the USA you would be offering "individualized instruction" rather than "private tuition". Note also that some schools offer "tutoring" programs in which a member of faculty meets with one or more students outside of class or after regular teaching hours. So the word "tutoring" has lately also become a recognized synonym for "individualized instruction" (In such cases, the teachers can be referred to as "tutors", which has resulted in "PRIVATE tutor" coming to mean either a tutor who is not a faculty member, OR any tutor who is paid by one set of parents to teach only their child.)
4) By DEFINITION, an adult can, indeed, be a "pupil", but long association of this term to young children has given it a CONNOTATION of referring to little ones. Since the middle of the 20th century, it has become more common to use the word "student". In the USA, for example, "pupil" is now slightly outmoded, even in their elementary schools.