This sounds like the beginning of an extremely bad joke.
Maybe climb through the window?
why isn't it domon? isn't the house the object of this sentence?
In the notes it says that nouns after prepositions (in this case "al") don't take the "n" ending. If I recall correctly, it's because that means it's an indirect object, and the "n" is only added to direct objects.
Yes, but "kiel la komencanto iras domon" could be an alternative (without "al la").
No, "house" is not the direct object. It is not the recipient of any action. The house is not being gone. Someone is going to the house. If Esperanto had more cases, it would be in the locative.
Why kiel instead of cu when it is a yes or no question?
Because it's not a yes or no question. It's a wh-question. Who, what, when, where, why, how.
So, a word for word translation would be, "how the beginner goes to the house"?
Yes, exactly – just like in most languages. The English construction with "to do" is really odd and unusual.