Answer to your first query : You need to make a written application or send email to HR and mention that as per employment agreement there is a termination clause either side to resign or terminate than Lieu period of 3 months in advance notice or basic pay has to be paid.
So you have full rights to ask for 3 months full payment. Otherwise you can make complaint in the labour commissioner or registrar and later on file complaint in the labour court.
Answer to your second query: No need to pay fees for past six years back here you can make written complaint to education department of your city that is called district forum and keep in cc to state education ministry as well.