The Director/Principal of my school likes me, so she let me self advertise on the school and hire students, but what about other schools? If I'm going to encourage self tough game development in other schools, would i need any special permission? (this "ecouragement" might involve hiring, speaches, self advertise, couching, etc).
What do you think it's a good way to start without looking "invasive" or "self-seeking"?
It depends on where in the world you are planning to conduct your business. Public schools are usually run by a school board which is part of county or state government in the USA. Private schools are typically run independently and make their own decisions. If you're speaking about other countries, you should know the structure of how the school system is governed and what level of autonomy each school principal or director has to make decisions without board approval. Even in private school, the headmaster might answer to a board of directors.
There's no one answer that fits every situation. You are inherently self-serving if you're both attending school and trying to sell something to the school. That doesn't mean it's bad, but you do need to be aware of the ethical concerns that will be raised when anyone governing schools looks at how decisions about your product are made.
This is a question for you to ask your current principal and whomever they answer to. It is not something we're going to know at CoFoundersLab.