1. b. you use queria because if you say "quise" it means you stopped wanting to learn ballet sometime in the past.
2. b. you use conoci' because you've already met the professor in the past
I believe it's Voy, because that is the "I" or "Yo" form of the word "ir", meaning "to go".
Where are you going to eat?
(I am going to) The restaurant.
¿Adónde vas para comer?
Voy al restaurante.
