Modals

Raising his hand in class, Rohit asks "Can I go to the washroom?", thinking "can" is the right word for asking permission. His teacher says the correct, traditional word is different. Who is right?

Vivaan writes "I must to finish my homework", thinking "to" needs to go before every verb, the way it does in "I want to finish". His sister says modal verbs are not followed by "to". Who is right?

Maanasi thinks "You must drink more water" and "You should drink more water" mean exactly the same thing, just with different words. Is Maanasi right?

Farah thinks "You mustn't come to the party" and "You don't have to come to the party" mean the same thing, since "must" and "have to" mean almost the same. Is Farah right?