1) see In this case when Takahashi-san says he is going to "see" May Grey, it doesn't mean he is going to take a look at her. "See" is used in various different ways. Basically, of course, it means perceive with your eyes, but you can also perceive with your mind and in that case,
what you're getting at." But also see is used mean visit or consult. So, you see a doctor when you need to find out if you're healthy or not.
2) You of all people. "You of all people" is a kind of thing people might say when they see a bit of irony in the situation. I think, in this case, Takahashi-san seemed very happy at the company and so May Grey is a bit surprised that he is considering leaving. You can use the phrase "you of all" or "that of all" with almost any other group when you see one individual or one unit that's for some ironic reason acting differently from the rest of the group.
"Here's the story" isn't, of course, fiction. He is telling what's actually happening. Some other phrases you could use in stead of "here's the story" would be, for example, "here' the skinny," which is very slangy. "Here's the set-up" is another way you could talk about what's happening.
5) I'd be in charge This conversation is rather interesting in the verb forms that Takahashi chooses, he switches back and forth from talking about things as if they're not quite settled yet. It's not decided yet. But then now and then he'll say something that sounds like he's trying to hold back but actually deepen his heart, he has decided to go. In this case, he says I'd be in charge of external relations, "I would be" in stead of "I will be." So, in this sentence anyway, he's showing that he hasn't completely decided yet.
6) tempting A couple of other words Takahashi could have used in stead of "tempting" are "inviting" or maybe "attractive" or even "compelling." All of them talk about the offer drawing him towards making a change. "Tempting" makes it sound like he is in two minds about whether he really wants to go or not. It's very tempting but on the other hand, he likes where he is, so he is kind of going back and forth and back and forth. If he'd say "inviting," it sounds much more like it's a very pleasant to move to the second place and he is not really thinking about staying. If he'd said "compelling," it sounds like the offer somehow pushing him to take it. Although it doesn't really say anything about whether he's fighting that push or not.