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In this scene Saiki partly explains it but on some older versions like the Volume Zero (one shot volume), Saiki explains some deeper kinds of telepathy. You can also differentiate sometimes on chapters where the illusion of Saiki from the other's point of view how he speaks with speech bubbles but when it's on a box/rectangle it's his telepathy or the real image of Saiki talking since he doesn't move his mouth (unless he is shocked when Nendou/Teruhashi/Satou does something he didn't expected.
There was that chapter where he got sick on pollens too so I guess he's not that omnipotent but maybe Saiki's being stubborn here as he always (implicitly brag) to his father that he never gets sick. When I read this arc before I was laughing because I can see how he doesn't believe he's having sea sickness.
Those were just some personal thoughts though I believe Saiki can be prideful to his powers despite him saying he's not abusing his powers lol ironically, he seems to be willing on taking risks when sweets are at stake.
(the name was out of topic but yeah I was a huge fan of DC)
The author didn't mention it ;;_;; also, since this is just a parallel world, the one who died here is a different Kusuo who has different lifestyle and different attitude, let's just say it's Kusuo A in Parallel World A who died.
This arc was during the time Kusuo went back to past during his sleep right? Then he mistakenly went to different futures and this scene was at the time where Kusuke created a time machine to bring Kusuo A to life but instead, he started world war 3. You can find different worlds here saikikusuo-no-psinan.wikia.com/wiki/Category:Worlds.
Also I doubt he can die from illness since Kusuo himself says he hasn't gotten colds since he was young while I'm not sure about poison but almost nothing can hurt nor pierce him (injection or blades).
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To answer your questions...
Q1. How come that no one ever questioned that saiki doesn't speak? Or why he speaks into their minds.
A1. Saiki doesn't speak, he uses telepathy (speaking through their minds) his reason for not speaking must be because he got used to it? It's said he was able to talk telepathetically when he's just a few months old.
Q2. How does he answer teacher's question if he don't speak?
A1. He uses telepathy and also since he's got all kinds of powers, he might have used Hynopsis and Telepathy on the same time so people don't realize he doesn't uses his mouth when he talks to them. You can see a list of his abilities in here saikikusuo-no-psinan.wikia.com/wiki/Saiki_Kusuo/Abilities.
Hello there. Sorry, been busy with some things. I've been busy with life that I forgot to update some things. Season 2 of Saiki K is coming up soon. As for the time, Asou-sensei is still spoiling us TT^TT maybe the time is gonna be released soon. I'll be translating his latest tweets via Saiki Wiki twitter account to keep you guys updated.
Kusuo1412 Talk Saiki Kusuo no Psi Nan 04:34,1/10/2017
I can see Metori getting gya gya gya over Kusuo and would ask him to be his friend (which he failed to do so with Aren and Riki) Chisato lol maybe she'd ask for a million bucks? XD
Shun would be the first to react thinking if Kusuo knows about DR and Akechi Touma.... funny considering the fact that Touma is a bit of obsessive towards Kusuo and he already suspect him to be an esper, he'd really ask many many questions to Kusuo. Kokomi would turn beat red if she knows Kusuo can hear her out of this world narcissistic thoughts. Riki I can say he doesn't really have any overreactions except for the part where Kusuo can't hear his thoughts maybe that will earn Kusuo an annoying grin from the latter. Here's my fav, Aren. I bet he'll be thrilled since his former Ibaraki Gang is named ESPER (blame Asou!) so I guess he'll ask Kusuo about gangs and all or ask how strong he is. Hairo would be very competitive and would start one sided competition on Kusuo and Chiyo... I bet she'll realize the reason why her love antics doesn't work on Kusuo 😂😂😂