1) The Core is probably one of the most protected parts of the island. Jenka even created the Labyrinth and brought the Gaudi there to defend it from invaders...
Jenka created the Labyrinth as far as we know. Placing the Gaudi there to defend it. So, how would humans or even robots be able to attempt to build a huge structure like the Core Base, which seems connected to the Waterway, which might influence the rest of the island?
It's possible the robots raided and killed the Gaudi to a point where they were able to use machines like Monster X and teleporters to transport great amounts of material and build the Core Base in secret... This would have occured during or after the war.
It's also possible that as in Noxid's mod, they carefully drilled their way there, somehow bypassing any protection such as the labyrinth. However, this too would have been a massive undertaking. Is there any evidence of such drilling and tunnels? If the Waterway could be those tunnels... Then why is the entrance an ancient looking cat head (as seen in Gunsmith's lair...)
It's possible humans were able to somehow reach the Labyrinth and peacefully deal with the Gaudi, allowing them to safely construct the Core Base... However, I get the feeling the Gaudi wouldn't allow such a large project to occur in their midst... Even if they don't remember their purpose, most seem quick to lash out upon intruders.
I still have my doubts about bypassing Jenka's Labyrinth, and building such an unknown sized Core Base at the heart of the island. Of course, if it was during the War, and Misery and Balrog helped out, then the Gaudi would pose no threat.
2) Monster X, a machine possessed by a giant cat. What was that machine's purpose? Who made it?
I am guessing Monster X is possessed by a giant cat spirit... [What better way can you explain the giant disappearing cat D:] As for who made it... if the humans or robots were able to secure the Labyrinth with or without the Gaudi's consent, then I guess that machine could have been used to transport or aid in the Core Base's development in some way.
Alternatively, if such a machine existed in Jenka's time, or if she controlled or made it; it could have helped construct the Labyrinth...
We don't consider technology and Jenka or Ballos. But... it's at least worth questioning. After all, magic could always be unexplainable technology, far beyond our own; rather than an existence seperate from science D: Just a possiblity.
If there was a big chasm there that Monster X crosses (as depicted in some art), why not just build a bridge? Unless the chasm acts as a natural defense to prevent crossing...
3) The teleporter in the Labyrinth. The common looking computer in Core Base.
Is this technology brought by the humans and robots? If so, then we should consider the location of the Labyrinth teleporters. Perhaps, materials from the Labyrinth could have been used for the construction of the Core Base? But, we've never seen anything but people and robots being transported using the teleporters. So, I have to question how much can be transported... If the technology is the new human/robots, then perhaps there could have been a large scale transportation device for transporting equipment? Or, if it was during the War, then Misery could have easily transported large amounts of materials...
As for the computer, if it is a recent machine; then it makes sense that the humans created the base somewhat recently. Before, during or after the war.
However, unsubstantiated; if Pixel merely used the only graphic associated with computers... rather than make a custom look for the computer... then it could be possible that the computer is much older.... A long shot of course, but I can't help but reach for it

[I've read that in the 3DS version the place looks even more technological?]
4) The waterway, and the house there.
The waterway. Was it created at the same time as the Labyrinth, by Jenka or someone around that time. Or was it created by the humans/robots during recent times through drilling? The strange exit of the Waterway is the same as the Gunsmith's lair. Could there be some connection between Gunsmith and that Waterway Cabin?
Having a house in the waterway lends support to the case that the Waterway could be a recent development... which supports the Core Base as a recent development.
Some questions to wonder about... how does the water flow naturally? Was the Core Base area before it was Core Base, an underwater zone? Or did the drilling end up flooding the area? Intentionally or unintentionally...
However, if the Waterway was created by Jenka or someone else, long ago... Then perhaps it was crafted along with the Labyrinth? (Like a natural defense against digging into that area. You could be flooded!)
Another thing: perhaps while functioning normally, the Waterway, is not so flooded? Only once Core Base's Core was stolen/damaged did the waters flood? So surviving the waterway was a much surer thing. (Sue washed up in Mimiga Village from the Waterway... presumbably somehow from the Plantation's connection?)
If the Lion head grate was an ancient thing, then that might support the Waterway being very old as well... Which might lend support to the Core Base existing long ago... As for the Cabin, there are always alternative routes that we cannot see on the maps; such as how Curly can pass by you, while you fight Monster X...
5) The computer in the first part of the Labyrinth.
An odd thing... The Labyrinth that Jenka made, has a computer controlled door? If the computer and door was created when Jenka (or someone) crafted the Labyrinth; then that supports the Core Base as being much older...
Its use? Perhaps it was originally the entrance to the Labyrinth? There is only the exit now, but perhaps there used to be another door..?
Alternatively, if all of the technology is a recent development... It seems more like a prison. But, why would the control for the prison exist in the same area? It seems like a very strange chamber... Unless it was once filled with water? Or if there once was a way out further up..? Was something kept there perhaps? Too strange...
There is also the repeated comments about the Labyrinth being a junkyard, or place where the islands trash ends up... Could the Labyrinth be more advanced than originally believed? If the floating island was once a magical (or partially technological) kingdom?
6) The huge change in look from Core Base to the rooms next to it.
Having the Core Base being such a drastic change in look from the Labyrinth might support the Core Base as being a recent development. But, then again.. if Core Base existed first, and Jenka's Labyrinth came later...
Still it is strange, for such a sudden change. If it was recent, then wouldn't there be more evidence of paths the creators would have used? The Gaudi nest path seems too overrun... Although there is that teleporter there...
If the Gaudi were held back from interefering with the builders, until something happened.... It might have forced the builders to hide in the Core Base, where they were slaughtered by a surprise attack from the Core...
That would support the recent Core Base theory.
But then again, Misery could have just teleported the Robots in the labyrinth and have the same thing occur. We know at least some cases of this occuring... It's even possible, the teleporters were brought down there thanks to Misery's teleporting...
Well, what about the case where the Core Base was already there... but humans/robots/scientists find that place?
7) The water the Core Base controls.
It would take quite some time to be able to shift the water properly, or maybe have the Core control the water level... Wouldn't it? That might lend support to its prior existence...
Though if the lion head gate was built long ago in order to control the water flow... if they were able to use that; it might not be too difficult to control the water...
8) The robots mission. The robots grave in Core Base.
Depending on the translation...
"The purpose of the mission...
it was the incredible-
slumbering power found on-
this floating island...
And it was also about the-
Demon Crown.
But you and I were different."
Was the incredible slumbering power the Core? Or was it Ballos?
On one hand, Quote and Curly seem to be given additional background on Ballos from a mysterious source. Is it information that only the player reads? Or is it memories or commands surfacing?
If the mission was the Core, as possibly evidenced by all of the robots that ended up in the Core Room dead...
It would make sense. As the information on Ballos only appears after successfully fighting the core...? A secondary objective?
However... Curly also ends up heading to Hell with you. And the information on the mission was from Curly... And the context of the mission includes the Demon Crown which is definitely related to Ballos. So... It could make sense if she was referring to Ballos, although the player might assume otherwise...
On the other hand, Booster and Kazuma were able to find out about the Core, whereas Ballos is never mentioned...
.... Confusing nonetheless...
IF The Core Base was built by humans recently, then undoubtably the target was the Core. However, if the Core Base was not... then perhaps Ballos was the target after all.
As for the robot grave... were a majority of the robots killer robots or peaceful robots...? Hmm...
9) Any dialogues that speak of the Core Base.
Almost nothing speaks about the Base itself, but there is plenty related to the Core and Labyrinth...
"You were thrown down here by the-
witch, Misery, too.
Poor little guy...
The island's junk is stored here.
Once you're in here, you'll never be-
able to get out..."
"Long, long ago, there lived a witch on this island. She's the one who constructed this labyrinth, in order to keep us Gaudis penned up."
"The one who sealed us up in here is a witch named Jenka. ... But it's not because she hates us or anything. We're here in this labyrinth in order to protect something. ..."
"This is the legendary-
Labyrinth.
I don't know who built it,-
but it must be for some kind-
of trash collection."
"You see, this island floats in the sky by the power of a certain organism, and I've recently discovered its location. This labyrinth. The organism lives somewhere within this very labyrinth. If one could destroy it, the island would cease to function. *chuckle* The doctor must really be in a rush, to send you into the very heart of this island."
"Are we still in the Labyrinth?
We can't pass through here.
We're going to have to keep-
moving forward."
"This is a strange place.
Was it some kind of base?"
"Blast door after blast door.
There's something behind all-
this crazy security.
Don't you think so, too?"
IF they could construct that Core Base down there, what would keep it from waking up and destroying them during the construction process?
10) Nurse Hasumi, Doctor Gero...
.......... Hmm...
They used to run a clinic in the Labyrinth. They aren't human... probably. [transformed I mean]
They take care of robots like Curly... and Gaudi as well... ..............................
Their clinic was haunted by a Balrog like Pooh ghost....
Are they from the ten years ago!? Or perhaps some kind of magical creature that has been supporting the Labyrinth for a long, long time? I have no idea...
11) Ghosts and spirits in the Labyrinth?
I blame Doctor Gero... o_o
Okay, perhaps the labyrinth has its share of angry spirits due to being a place where enemies are dumped down there to rot and die. >< Perhaps the enviroment has a stronger connection to the spiritual or something... Whatever the Core is, it might fuel and attract things like that?
Perhaps that might give some insight into Jenka's magic...?
12) The physical age of the Core Base.
The Base looks old. Could ten or so years turn that base to that state? Sounds likely...
Could it be as old as Jenka though? Ancient technology in anime lasts a long time o_o
The real question is the symbol on the walls as seen in Core Base. Find it anywhere else and you might have your answer...