I have worked for the same company for over 5 years, paid tax, and they have renewed my Visa and Work Permit each year. If I was made redundant, what compensation should I be entitled to?

See section 118 of the Labor Protection Act here. http://thailawonline.com/en/thai-laws/laws-of-thailand/105-labor-protection-act-be-2541-1998-.html

" An employee who has worked consecutively for at least three years but less than six years shall be paid basic pay for not less than 180 days at the most recent rate of basic pay received by him or not less than the basic pay for work performed in the last 180 days in respect of an employee who is rewarded on the basis of his output;"

Does this apply if a school renews a series of 1 year contracts for 5 years and chooses to not give a new contract for the 6th year?

I believe so, but will also depend on the type of school and if there is time between the contracts.

The labour office can advise you of your rights and will lay a claim with the employer on your behalf if they don't want to pay. The labour office will even go to court on your behalf if needed, but most employers don't let it come that far.

You believe so but not sure. Ok.

Maybe private schools are not bound to this or maybe they lie and assume nobody will discover the truth.

As said, contact the labour office.