Foreman says that he should "ビブラートに包んで which means to "wrap something in vibrato"
which is a misquotation of the metaphor "オブラートに包んで": to wrap something in Oblate (these two words are both loan words and sound similar which is why jp people mix them up)
Oblate: a thin paper used to wrap medicine - figuratively, to sugarcoat your words/say something in an indirect way.
Vibrato: an opera technique where your voice goes up and down in quick succession.
In the next panel, the original text says: "what will you do if your voice wavers?" as in, "you still need the vibrato though" so I've adapted the joke to imply that, "yes, but your voice still needs to be intensive"
They play the "would you rather game" with the choices being either having immortality or dying, and Foreman brings up the question about what happens if your body dies. The foreigner says "思念体" which means "ideology" and it makes everyone confused. This is a homophone for "死ねん体" which means invincibility.
The joke is that Japanese learners don't have perfect pitch which makes miscommunication like this sometimes happen, and he fixes this by clarifying what he meant in the next panel.
The final joke is a remark said by the younger guy. He says 平地でも滑ることあるんだ: "even on flat ground, you can slip", a proverb which means that when you think you're safe, you can still make mistakes. The foreigner's Japanese is amazing but he still makes little mistakes like these. A pun; he's a mountaineer yet still trips over the small hills. I was going to leave it as that but unfortunately in the same panel, foreman tells is that it was a dad-joke. So I hope you enjoyed the "Ain't No Mountain high Enough" song reference.
and because of this interaction, at the end of the chapter he finds his niche as a comedian.