Translation:My little sister is very quiet at school.
You'd use younger sister in this case, young sister is incorrect.
We use the comparison form of young and old when referring to siblings because we are comparing their age to our own and stating that they are in fact younger or older than us.
Let's use the example that a man is 90 and his sister is 80. His sister is old, but in comparison to the man, she is younger than him. You couldn't refer to her as young. But you could still say she's younger than him, or that she's his younger sister.
I believe shizuka is an adjective, just like quiet is an adjective, but in "shizuka ni shite," to be quiet, the "to be" part is definitely a verb.
Is there anything wrong with my sentence? 妹は学校でとても静かです。Duolingo didn't accept it. I'm guessing it doesn't work that well when I use my computer keyboard, rather than picking the options that Duolingo has available.