Nichijou/WMG

Everything is happening exactly as you see it.
All the What Do You Mean It's Not Awesome? scenes and Imagine Spots are actually happening. Things really slow down and get dramatic, Misato is really shooting Sasahara (and he is really walking away injury free), people actually turn blue with shock or experience color failure, etc. The rules of the world in Nichijou work slightly differently than in our world and this really all is just everyday life.
 * This was actually what I assumed when I started watching the show. And considering the odd stuff that happens, it wouldn't surprise me in the slightest if this were true.

Professor isn't really a child. She's an adult who de-aged herself.
Nano is simply taking care of her until she either finds a way to return to her original age or returns via The Slow Path.
 * Well, for an adult, she sure is childish.
 * Adult Childish?
 * My thought was The Mind Is a Plaything of The Body.
 * This is what I originally assumed, but it is a comedy anime with a lot of crazy shit going on, so I wouldn't be surprised if she really was eight years old.

This show is the world seen through Osaka's eyes.
For starters, she's Nano and Chiyo is the Professor. Mai is what she imagines she'd look like if she were Sakaki.
 * I don't buy the Azumanga Daioh comparisons. For starters, who has the pigtails in this analogy?
 * Mio.
 * Who's Sakaki?
 * Mai is who she imagines Sakaki would be like if Sakaki had been born into her body.
 * Then Yukko is Nichichou's Tomo.
 * So Sakamoto is Chiyo's Dad?
 * It still leaves Yomi, Kagura and Kaorin out. And Mio's role is still unclear.

The show takes place in the Gag Manga Biyori-verse
In the third season of Gag, a robot shows up who acts very much like a human, yet has a lot of weird functions, just like Nano. No, they aren't one and the same (the robot is jerkish, Nano isn't et.c.), and the Professor didn't create the other robot, but this suggests a similar technology branch in the verse(s) of these shows.
 * Also, strange and awesome things happen on a daily basis in both of these anime.

Nano was not created by the Professor, but by someone who needed to oversee her safety without being there themselves.
Even though she knows enough to modify Nano without her knowledge (Nano may not have shipped with the key), she doesn't have the heart to reverse-engineer her to make her a servant instead of the other way around.

Half of the show is actually about Yukko.
Recent episodes are putting an increased amount of focus on the abuse she takes daily. Granted, it's mostly her fault due to her not controlling her Hot-Blooded and Book Dumb tendencies. She also stars alongside Nano in the ending sequence.
 * The second opening also gives Yuuko equal billing with Nano, either presenting them together or before either Mio or Mai.

"Hakase" is actually the Professor's real name
Hakase (博士), meaning "expert" or "learned one", is an honorary way to address doctors or professors. Spelled with different kanji (葉加瀬) it's also a fairly common Japanese family name. Since in the manga "Hakase" always gets spelled in hiragana (はかせ), it could even be the girl's given name, although that is a bit unusual. In any case, it could well be that Nano is calling her by a proper name rather than a title.
 * Following this thread: Hakase dresses in her lab coat and designs robots because, due to her name, she thinks she is a Professor.
 * The same play on words occurs in Mahou Sensei Negima with the character Hakase Satomi (葉加瀬 聡美), who also is an expert in robotics. Coincidence?
 * Except that in that case, "Hakase" is being used as her surname, not given name. (Japanese order, people.) The pun is still there, certainly, but it's not exactly the same scenario.
 * Since it's common in Japan to call people by their family name, this is not really an issue.
 * Nakamura's analysis of the name in episode 24 casts doubt on this theory, unless she's actually named after her occupation.

"Helvetica Standard" is the future of the Nichijou universe
Professor grows up and makes more awesome/weird inventions, so either Reapers, angels, tengu and such like are said inventions, or it's that normal people can see them now. All the while she and Nano (who went through some cosmetic changes) sit in a cafe and complain that coffee isn't "coffee-coffee" enough.
 * But "Helvetica Standard" already exists in Nichijou as a manga series...
 * Maybe Professor builds her new inventions after characters of "Helvetica Standard" ?

The Professor based Nano's design on someone she knows
Either consciously or subconsciously, the Professor designed Nano's appearance on someone she's met. Hilarity shall ensue when Nano and her "original" meet up.

Nano had a Restraining Bolt about the windup key
The Professor installed a Restraining Bolt on Nano's programming to prevent her to remove the key on her back herself. That is why she kept asking the Professor to remove it, although Mio's sister showed that it could be easily taken off.
 * Confirmed in the last episode. Anyone except Nano can remove the key. The Professor does mention there's no such restriction about putting it back on.

The reason why the show is called Nichijou...
...even though nothing on the show is really "ordinary" is because it's partially based off of things in the MLIA site. Seriously, go here, and you'll start thinking that either the people on the site can't define "average" or that maybe life isn't as normal as we think it is...

The real reason why the show is called Nichijou...
It's about Nano trying to get an ordinary life, and the things around her are anything but.
 * The final episode has Nano Title Drop it in just that context.

Nano was created in the future by the Professor to take care of her younger past self.
And Nano doesn't realize this, which is why she keeps asking the 8-year-old Professor to remove her key when she only refuses. She doesn't refuse because she doesn't want to remove it, she refuses because she doesn't know how! This may not explain how she is still able to install small features into Nano at that age however.
 * Does this make Mio's sister a potential Spanner in The Works given how she once just pulled it off?
 * She definitely knows how - she even goes through the trouble to install a small one in Nano is the final episode(after she built it thrice).

Igo-Soccer is someone's practical joke Gone Horribly Right
Someone invented Igo Soccer as a nonsense game just to see who he could trick in to playing it... then it snowballed.

Misato and Mio will have a kendo fight for Sasahara's affections.
In the course of it, Misato will bring out her weapons, and Mio will unleash the Wood Cubes. And it will be awesome.

Kana Nakamura is the original owner of Sakamoto-san.
She wasn't just chasing after a cat that looked like him because it would provide her more information about Nano's machinery!
 * Confirmed via Early-Bird Cameo in episode 3.
 * How convenient that Sakamoto was covered with the Henoheno mask in episode 24 so she wouldn't recognize him the first time.
 * Episode 26 has her see him sans mask - and yep, he's her runaway kitty.

Mai isn't a Troll, she just has No Social Skills.
A lot of Mai's more trollish actions make as much, if not more sense, when you consider Hanlon's Razor -- "Never attribute to malice that which is adequately explained by stupidity."

The Principal is related to the Professor.
Episode 26 noted,. If this isn't just mere coincidence, it explains a lot about how the Shinonome Lab is able to get by when nobody is employed, and also how the Professor got Nano into school in the first place.

Yuuko has a learning disability.
In those few times she actually bears down to study, she is unable to stay focused for more than 10 seconds. It is only through a miracle that she scores as high as 80 on a quiz legitimately.
 * Additionally, she scored 80 legitimately and it was still a below average grade.

The Professor is the granddaughter of Dr. Gero.
She can make a android and she gave Sakamoto, of all things, a RED scarf.

There's only so much Awesome go to around.
In the Nichijou-verse, there's a set amount of Awesomeness given to each individual or group of individuals. This is why so many characters can do such over the top things like turn the three-second rule into a scene from Dragon Ball, or a tsundere express her tsun-tsun side with heavy weaponry. This is also why the scenes featuring Shinonome Laboratories are so mundane by comparison. As a mad scientist child, wind up robot, and talking cat, the Shinonome are so bizarre by default that there's no very little extra Awesome to go towards having those same kinds of scenes.

The oddball stuff in the show is all in Yuuko's head.
Think about it. All the really out there, or extremely exaggerated stuff in the show (Principal wrestling Deer, Mio's assault on the cop, the Wooden Cubes' origin, etc) has Yuuko present or in the case of the cube origin, sees the cubes change. What if, considering the fact that she doesn't seem to really care about school, these events are her daydreaming? Maybe their lives really are ordinary, but we see the show through Yuuko, and we see what she thinks.
 * It could also be schizophrenia, in some cases.
 * Nano could possibly be just a normal girl, but Yuuko envisions the screw to make things more interesting. She probably also daydreams all the stuff that she's not in either, like the Igo Soccer stuff and the Professor & Nano's home life. Clearly, this would mean Sakamoto wouldn't be able to actually talk.

The Professor isn't the real Professor Shinonome.
This is a theory that combines a few WMG here. The Professor's father is the real professor Shinonome and owner of the lab Nano and Professor live in. Being a bit of an Absent-Minded Professor, he named the Professor "Professor" / "Hakase" in hopes that he would be able to teach her everything he knew about science. He homeschooled the Professor to do so, giving the reason why she doesn't go to school. Eventually, Professor Shinonome had to leave to go do science somewhere, and he built Nano to take care of the Professor while he's gone. With her father abroad, the Professor believes that she has inherited the title and responsibilities of being Professor Shinonome, but the only real things she remembers her father doing was creating Nano, who she knows how to modify due to her homeschooling. So, when Nano says that Professor created her, the Professor agrees, because Professor Shinonome did, not our Professor.

The Professor's real name is Kurisu Makise
And the world has an over 9000% divergence from Steins Gate. No, seriously. Blue eyes, red hair, scientific achievements at a young age...

The Professor is actually the Doctor
She's far more intelligent than any human has any right to be and her name can be translated as "Doctor." Logically, the Doctor regenerated as a small girl in Japan and decided to settle down for a while because, as a small girl, settling down for a while seemed like a good idea. Nano is her companion and her TARDIS is hidden somewhere in the house. To elaborate on her intelligence, she gave a cat the ability to talk. If we're being realistic (even though there's very little of that here), cats are not equipped with the circuitry to even allow the abstractions necessary for coherent conversation, meaning that she gave the cat sentience with a scarf that she built in 5 minutes. Of course, there's also the Ridculously Human Robot Nano, who she apparently made at the age of 7.

The Professor is actually Agatha Heterodyne
After all, how many long blonde haired girls capable of building Nano at age 7 are there? They even both have officious talking cats. Perhaps she dimension-transported herself, and later returned with amnesia to become the Agatha we know. Maybe the Professor is an alternate-universe version of Agatha. Whatever, if you DNA tested the pair, you'd find a match.

The Professor (And, by extension, Nano) are employed by a weapons manufacturer.
This explains how the Professor and Nano are able to live in a house on their own without any apparent employment or adults, nor Professor going to school. The Professor is obviously a child prodigy who developed a robot on her own, and a robotics company tuned in and scooped her up, giving her and Nano the funding needed to live on their own. They are the ones urging Professor to give Nano those "alterations" and, with complete data on how they work, figure out how to implement them in their own robots(Developed from Nano's design) for more sinister purposes. The gun(though airsoft) in particular is a glaring stab at this idea.

This is your Haruhi on drugs.
Any questions?

The Professor was originally intended to be one of the Schoolgirls.
Take a look at the Nichijou's main page pic, which is of the manga's first volume cover, and you'll see a strawberry blond, blue-eyed girl who stands out from the rest. The only character in the series with that same hair and eye combination is the Professor. This lends itself to one of two possibilities. The first, noted above, is that Keiichi Arawi originally intended to make the Professor a teenager instead of a child, but changed gears somewhere between conception and publishing to make it the Professor's robot Nano.

The blue-eyed girl on the first volume cover is an early version of Yuuko
The second theory noted above. The character with the blue eyes was the original design for Yuuko and, again, Arawi changed gears before proper publication and instead gave that design to the Professor.

Sasahara knows about Tachibana's crush
Sasahara is actually Genre Savvy enough to notice Tachibana's tsundere attitude, and he's also aware her attacks are non-lethal. He just trolls her into embarrassment because it amuses him.
 * Or he tries to let Tachibana know he knows about her crush on him, out of mutual affection.