Is this a funny joke?

[Dee3]

Hi. I have some questions about the following sentences.

Marcus is suddenly beside me. I introduce him to Dewey and Hollis. Dewey shakes his hand and then continues to mouth-breathe and look distracted. Hollis promptly asks Marcus where he lives and what he does for a living. (Then they find an excuse to move on to more worthy guests.)

1) shake one's hand // Is this different from "shake hands?"
2) mouth-breathe // the meaning?

Marcus raises his eyebrows. "Dewey, huh?"
"Yeah."
"Dooo heee have a stick up his ass or what?"

3) The last sentence looks like a joke because it makes "I" (the heroine) laugh. But I can't figure out what's funny. Could you explain it?

Thank you.

[Mister Micawber]

1) shake one's hand // Is this different from "shake hands?"-- No, they are the same.

2) mouth-breathe // the meaning?-- It suggests that he may be upset (from some earlier text)...or he may just have the habit.

3) The last sentence looks like a joke because it makes "I" (the heroine) laugh. But I can't figure out what's funny. Could you explain it?- 'Do he' sounds like 'Dewey' and ' Do (does) he have a stick up his ass?' means the person looks stiff and uncomfortable. She could have laughed at either or both.
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Native English teacher at Mister Micawber's

[Dee3]

Thank you for your reply, Mister Micawber!
As for 1), at the time when I posted the question, I had thought that Dewey shakes his (= Dewey's) hand like waving his own hand as if to say "Hi." (Impossible?) Since it takes two hands to do that action (handshake), somehow I thought "hand" should be always plural for that matter. But your comment has made me realize that Dewey shakes Marcus' hand. (Right?) Here are some handshake-related sentences I've just made up. Please check them.

1) We (Tom and I) shook hands.
2) We shook hands with each other.
3) I shook hands with him.
4) I shook Tom's hand.

[Mister Micawber]

Yes, those are all fine and natural for handshaking.
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Native English teacher at Mister Micawber's