In which cases can you ask a question without using auxiliary verbs?

Questions with the verbs "to be" and "to have ~ed" don't use do. Is he a teacher? Has he taught math? Neither do questions where the subject is a question word ("who", "which", "what", etc.) Who ate my pizza? (but "Why did you eat my pizza?" -- "you" is the subject) Which bag is yours? What is your favorite kind of ice cream? How many cats live in your home? (but "How many cats do you own?") In very casual speech in questions directed to "you", you can drop the subject and do/does/did. Do you want to go to the beach? = Wanna (= want to) go to the beach? Do you think he's happy? = Think he's happy?

Questions with the verbs "to be" and "to have ~ed" don't use do.

Is he a teacher?
Has he taught math?

Neither do questions where the subject is a question word ("who", "which", "what", etc.)

Who ate my pizza? (but "Why did you eat my pizza?" -- "you" is the subject)
Which bag is yours?
What is your favorite kind of ice cream?
How many cats live in your home? (but "How many cats do you own?")

In very casual speech in questions directed to "you", you can drop the subject and do/does/did.

Do you want to go to the beach? = Wanna (= want to) go to the beach?
Do you think he's happy? = Think he's happy?