...how does Tranquility look? is she cute?

Yeah, she's reasonably cute. Why does he ask?

He does not hand the atlas to the Chairman. He walks over, places his hand around Tranquility to rest on her shoulder, and guides her to the back of the couch, on which he one-handedly props open the atlas.

"Ah, Tranquility-chan, please grant me the honor of showing you." He finds the world map, and indicates the location of San Francisco.

How is everyone else reacting to this?

Kozue is honestly not at all jealous to see the guy she occasionally messes around with doing that, everyone knows Touga's a playboy. 

Who is this fucking WHORE? Who is this fucking TRAMP, this SLATTERN already trying to seduce Onii-sama, mere moments after laying eyes on him?

No one else seems particularly fazed.

She is not particularly excited by this sudden invasion of personal space by someone she just met a few minutes ago, and attempts to move out of his grasp subtly and politely (though she does initially jolt when his grasp first lays on her shoulders). Nonetheless, she'll look at the map, and the location he points to.

"Yes, that is where the Golden Capital is. And Ohtori-san said we were in Japan... where is that?" (Finally, a chance to use the honorific and correct her previous mistake!)

She scans her eyes around the rest of the map while she waits for a response. Nothing has the name she remembers, and the map is rotated 90 degrees, and there's lines criss-crossing the map that kind of correspond to some natural features she knows about but often don't correspond to anything she remembers at all.

Yeah, this is not Melodia. What the actual fuck.

He withdraws his hand, but not his intentions. All in good time.

He indicates the location of Japan.

She will remain slightly suspicious but appreciative of the immediate removal of his hand.

Oh interesting. She's never had the chance to visit that particular island chain, but it has a decent number of people on it. She wonders whether this will count as "visiting" it now - this is probably going to be a lot different from the way it is in Melodia.

"Thank you, that clarifies... a lot. And also very little."

She's nervous about this next part, but it has to be done. If she is actually going crazy, letting others know is the fastest way to get back to reality. And if she's not crazy... who knows, maybe appearing in alternate worlds is a regular occurrence here, or maybe she missed some obvious thing that made her jump to conclusions.

"...so. I might be insane, or missing something obvious, but is people appearing from other worlds a normal occurrence here?"

"No."

(A-ko would like to make a joke, but she is not a sufficiently diagetic character to interact with Tranquility or anyone here! Being a literary device is fun, but sometimes it really blows.)

Touga puts on a well-practiced bedside manner and speaks with concern in his voice. "Are you feeling disoriented or feverish, Tranquility-chan?" He snaps his fingers and snarls at himself. "Of course. You washed up unconscious on the beach, what was I thinking?" He turns to a green-haired boy his age. "Saionji. Please retrieve a towel and blanket, this girl needs to be dried off and kept warm."

Saionji doesn't move.

A girl who looks to be the oldest student, around 18, takes the prerogative to get a blanket and a towel, and comes back.

She speaks to Tranquility. "I'm Ohtori Kanae. If you'd like to change out of your wet clothes, I'm sure one of us will be the same size."

"Thank you, Kanae-san."

He takes the towel from her, and motions at Tranquility in a way that could either mean 'Here, take this towel' or 'Here, let me dry you off.'