Mao's plan to "rule the world" via job climbing at MgRonald's seems funny, until you realize there have been business tycoons who started from the ground up. Part-time employee, full-time employee, manager, corporate executive, management vice-president, management president, corporate executive officer, chairman of the Board. And this is just business; by the time he gets to that level, he'll have learned how Earth society, politics and military works. Maou is an exception to only Humans Advance Swiftly; he's Long-Lived, he's patient. Give him a century or two to himself, and he really would be a rising power on Earth. (Look at Vandal Savage!)
It makes even more sense after you get to know that he wasn't the overlord forever - he began as a simple trooper in the demonic army and managed to become the overlord the same way he plans to get back now.
The above idea is also the reason why Maou seems to easily adjust to a simple life, compared to most overlords, who would probably not last a month or two living an impoverished life. He's simply back to his roots, and has not forgotten how he rose to the top.
In Episode 7, Lucifer finds Emi's bottle of Holy Vitamin B and jokes that if she keeps drinking stuff like that, she might get a heatstroke like Ashiya. Except, he knew what the bottle actually contained and Alciel had too much of it, thanks to Suzuno. It was a hint that they already knew who Suzuno was.
And because Lucifer knew what it contained, how else could have Olba known about it?
Following defeating Lucifer and saving the citizenry of Tokyo, the heroes' attention is drawn to Olba, buried under the rubble he caused to fall. One of them mentions that the police will identify him as participating in a robbery and arrest him soon. However, why didn't they take into account that he or Lucifer could have used magic to disguise themselves, considering they had seen both use magic? This comes back to bite them in the ass later when not only does Olba not get arrested, he returns to cause more trouble by assisting Sariel.
Even worse in the Light Novels as Olba escapes back to Ente Isla and is one of the main villains of the later Light Novels, stirring up trouble in the demon realms and with The Azure Emperor.
In episode 6 Mao and co go to an old school building that is apparently a supernatural hotspot, thinking that it might have a link to Ente Isla. Chiho explains that it was closed because several students vanished there without a trace, as if spirited away. Lucifer explains that places where Gates have opened in the past are often subject to supernatural phenomena. If they're right and a Gate did open there, then the missing students were likely pulled to Ente Isla or some other parallel dimension and never found a way to come back.
School officials likely wouldn't close the building down if the missing students had turned up unharmed somewhere. In other words, a group of kids wound up in the middle of a Trapped in Another World adventure in Ente Isla, but they never came back and never got to see their loved ones on earth again.
In the light novels, he made it back to Enta Isla and set about causing a civil war in the demon world.
Did the owner of the scam company in episode 13 look weird to anyone? Considering his method of dress, the smoking, and how he wanted to "take it outside" when Maou tried to argue with him...that business may have very well been a business front for the Yakuza.
In LN#4, Maou and gang go to a beach shop run by Amane Ooguro. A laid back woman, at one point she warns/teases Maou not to think about flirting with any of her scantily clad female customers. Later though, it turns out Amane is an aspect of Binah, a spirit protector of the Earth, and runs Ooguro-ya as a sort of Spirited Away halfway house for human spirits. She's not only warning Maou not to chase after the young women that come to her store because of Rule of Funny but because the customers are spirits and technically dead people.