Many organizations do not have proper processes in place and, therefore, do not bother to give out appointment letters. Their attitude is: you are getting a salary properly, so what's your problem? Others deliberately do not give it, thinking that it protects them in case of any adverse action by the employee (again, it's a fallacy).
So you need to gently ask for it without rocking the boat. If you take legal action, you will be promptly out of a job. One way is to say you are applying for a loan and they want the document.
About the notice period, schools do not come under standing orders; otherwise, you can always follow the notice period stated in it. However, it's a good idea to follow that as a guideline, to give one month's notice.