A: "Are you kidding?"
B: "Why should I lie to you?"

B imples that he is not lying. Let's see how that is implied. Let's say A answers the rhoetorical question.
"Hmm I can't think of a reason why you should lie to me."

So let's see.

P1: If I lie to you, I have a reason to lie to you.
P2: I do not have a reason to lie to you.

C: Therefore I am not lying to you.

Hmm... although the reasoning is correct, the first premise seems weird. Let's try another.

P1: If I have a reason to lie to you, I would lie to you.
P2: I do not have a reason to lie to you.

C: Therefore I am not lying to you.

Now the premise makes more sense, but the argument is invalid.
The reasoning for "How would I know?" and "What to do?" is quite similar. They are likely invalid too.