Hi Sasha!
The first one is correct and polite. đđ "May I be excused from class?"
"Can/May I go out" are both used. Usually used when one literally wants to go "out." "Hey mom can/may I go out now?"
The old can/may debate is not so strict anymore. However, formally speaking "may" has been seen as more polite. "Can" as a verb alludes to having the ability to do a particular thing. But they both refer to "possibility."
I've read a saying in English: "You may use "can" if you wish, and you can use "may" if it makes you feel better."
Take good care!