There are two questions:
When do our classes finish on Tuesday?
What makes him feel amazed?
I can't wrap my mind around why does the first question have "do" as an auxiliary verb, but the other doesn't have any. Or is "makes" one?
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Here is the principle:
When what, who, which or whose is the subject or part of the subject, we do not use the auxiliary. We use the word order subject + verb.
You can see this if you look at the statement versions of your questions:
Question 1: When do our classes finish on Tuesday?
Statement 1: Our classes finish at ___ o'clock on Tuesday.
Question 2: What makes him feel amazed?
Statement 2: ___ makes him feel amazed.
You can see that in question 1, we're not asking about the subject of the statement. In question 2, we are asking about the subject of the statement.