“School” is being used as a general term: grade or secondary school, public or private and buxibans.
“Larger chains” are also just buxibans. A buxiban, large or small, is just any business that offers courses on just about any subject. Not sure how much real help any of them give on settling in.
Yes, you can work legally for more than one employer as long as each employer secures you a work permit and you add them to your ARC. The total combined hours as applied for on your work permits can’t exceed 32/week.