English (USA) students would ask: Can I go to the bathroom? May I go to the bathroom? Can I go... ? is incorrect English because the person is asking if they are physically capable of going. Teachers will often respond with, "I don't know, can you? They are trying to remind students that they have posed the question incorrectly. But it is spoken this way all the time. May I go...? is correct because one is asking permission.

It's something that depends strongly on local culture. In any one school, you are likley to find the phrase used has evolved to some short code format. . Yes, "May I go out" sounds odd to me as it does not state the prupose, but asks for general permission to leave. . I have not been in school for some time, so not sure what is used now. Also, it might not be phrased as a question, but more as a statement, that gets acknowledgement. What age? I need to go to the bathroom. I need to make a comfort stop. May I be excused to go to the bathroom. Could I be excused to go to the bathroom. I have an urgent need to visit the bathroom. I need to take a bathroom break.

It's a little formal, but it's a native expression that a child or student would say. If an adult said it, it might sound sarcastic -- like the adult was imitating a child.