Can someone tell me to what do I need Black Stones Spell (the one you can cast only in the cat form)??

You have to know the levitation spell, you can learn it in the wizard tower. I don't remember if you have to be invited by him (via owl post) or you can just follow him there though

It depends on how far in game you are

You can buy it in the magical typewriter room (like you can buy broom)

You have to choose Achim and tap the action button when he is nearly someone who you want to go on the back of wheelchair