Everything in Haruhi-chan is...mostly canonRelated to the below, but not as detailed; everything in both series is canon, but is subject to Broad Strokes. Most of the Rule of Cool and Rule of Funny is edited out, but things like Haruhi's Charles Atlas Super Power dodgeball game in episode 5, Haruhi's scavenger hunt in episode 7, Yuki's omnipresent geekiness, and even, perhaps, the resurrected chibi Asakura, are canon to the orignal series, but Played for Laughs in this version of the series. Churuya-san is mostly not canon, although things like her love of smoked cheese and perhaps Ashakura's crush on Kyon might be canon.
Mikuru-chan is Jesus.She was put on a cross by Haruhi. And then there's the whole time travel thing. Therefore, she was put on the cross for our sins and the sins of the future.
Mori-san is the Chuck Norris of the Haruhi world.She freaking jumps into school buildings and causes craters when she skydives, and remains unscathed! Maybe Haruhi-chan wanted to give a human godlike powers to match her own.
Tsuruya is an Airbender.As proven in episode 17. Her symbol is a Crane, which is a bird, which is a symbol of wind!
Kyon is The FlashAs proven in Nyoron episode 2.
Yuki's doujin game also has a Taniguchi, Kunikida, and Computer President route.Yuki is aware that Kyon is the LOATS. She may have even put Shamisen as a route!
EVERYTHING in both series is canonNagato's character development eventually turns her into a full-on boke yuri/yaoi fangirl. And she moves to videogames, because, well, she's read every book available already. Asakura does return as a chibi, and character development makes her fall in love with Kyon (and her reduced size makes her develop a new catchphrase, "not possible"). Meanwhile, Haruhi's powers mess with Tsuruya, first turning her into a superpowered badass, then into a chibi ("I wish she was even cuter!" were probably Haruhi's words). Who lives with Kyon temporarily, at least until some cure is found, and ends up kind of disliked by everyone. Understandable, because after a few months of whining about smoked cheese and going "nyoro~n" every 20 seconds, even saints would get annoyed. Nagato temporarily rewrites reality to allow a silent mode for her, too. Meanwhile, chibi-Asakura's existance has been revealed, and of course Kyon is none too pleased with either her return or her attentions. Mikuru... is just having a bad week, never mind that.
Yuki's personality in Haruhi-chan is exactly the same as in the main seriesThe only difference here being that here she has no filter of any kind as opposed to the main series where she's so stifled.
As we've established that Mikuru is Jesus, and from there it's a logical extension that Mikuru is Haruhi, and she's a Time Lord by virtue of being a time traveller, Mikuru is also Shinji Ikari.In the words of Kinky Friedman, "why the hell not".
The series takes place in another Alternate Universe created by YukiA good chunk of the series is about Yuki, Ashakura, Mr.Kimidori. Its not too hard to believe that Yuki wanted to live in a world where she got along with her Alien peers, as seen in The Disappearance with Asakura. Besides that Yuki has more defining emotions and wants than her actual persona.
Haruhi-chan takes place in a universe created by Haruhi after watching Lucky Staran actual crossover with Lucky Star seemed was to unrealistic for her so she just made everyone in her world chibi, gave herself Konata's voice and made Yuki an Otaku.