Learn Swedish: Känner du honom?

han means "he" (the subject doing the action), while honom means "him" (the object receiving the action). Because "he" is the person being known, he is the object of the sentence, so we use honom.

How is honom usually pronounced in everyday speech?

In spoken Swedish, the "h" is very often dropped, and the first "o" sounds like an "aw" sound. You will frequently hear it pronounced as "å-nom", making the sentence sound a bit like "Känner du ånom?" when spoken quickly.

Could I use vet instead of känner to say "know" here?

No, Swedish has different words for "to know". You use känner when you are familiar with a person or a place. You use vet when you know a fact. Since you are asking if they know a person (honom), you must use känner.

Why is there no word for "Do" in this question?

Swedish doesn't use helper words like "do" to form questions. Instead, it just swaps the order of the subject and the verb. Starting with the verb Känner instead of the subject du is what turns the statement into a yes/no question.