How did Sora, Donald, and Goofy end up at the grassy area outside of Castle Oblivion? Did they go there simply by Gummi Ship? Or were they sent there by being sucked into a Corridor of Darkness?
At the end of KH, during Sora and Kairi goodbye, you can see a road behind Sora where the Door was. Maybe they just followed that road and Marluxia lead them there like shown in the beggining of CoM.
What was written on the letter that Pluto was holding in the beginning of Kingdom Hearts Chain of Memories? It's obviously not the same one from the end of Kingdom Hearts II since Mickey wrote that at the end of Kingdom Hearts coded.
One of the questions that I've always had and no one knows the answer and it seems we'll never know.
In the first Kingdom Hearts, how did Mickey know about Sora and Riku, and how did he know that they got a Keyblade? Yen Sid has the ability to sort of sense things and people in other realms and worlds, so did Yen Sid tell Mickey about them?
I don't think Mickey knew about Sora and Riku. He actually just asked Donald and Goofy to find the Key and they decided to go to Traverse Town and begin their search there. Happy coincidence maybe?
How did Axel and Saïx know about the Chamber of Waking?
Xemnas gave direct orders do Saïx and Axel to search for it in Castle Oblivion as seen in 358/2 days.
Why does Xemnas and Ansem, Seeker of Darkness have orange eyes instead of yellow?
Exposure to darkness turn the eyes to an orange color, like Ansem The Wise.
It was stated that the pieces of light of the χ-blade were split into seven pieces to protect the number of pure hearts in the world. This is referring to the Seven Princesses of Heart. The Seven Princesses of Heart didn't exist way back in the Age of Fairy Tales though, so how did the χ-blade know that they were going to be born in the future?
Like you said, the χ-Blade is from the period of χ and Union χ. Since Master of the Masters and the Foretellers know the future it might be related to them. We just need to wait for Union χ ending and a lot of questions form the mitology might be answered.
Braig says he has his hands full with his own plans. What are these plans and what is he off doing?
This isn't much of an answer but more like a guess. The way Braig/Xigbar acts and talks remind me a lot of Master of Masters, so maybe Braig turns out to be him and that's what he is refering to as his own plans...
In the Xehanort's Reports, it is stated that when the χ-blade is forged, then the real Kingdom Hearts will appear. So, we can assume that the Kingdom Hearts we see in Kingdom Hearts Birth by Sleep is the real Kingdom Hearts that came out of the darkness it was lost to way back in the Age of Fairy Tales, because that was around the time that the χ-blade was being forged. But, in Kingdom Hearts Dream Drop Distance, Master Xehanort says "...the only real Kingdom Hearts was swallowed by the darkness, never to surface again." So, if the real Kingdom Hearts never surfaced again after it was swallowed by the darkness in the Age of Fairy Tales, then what Kingdom Hearts is the one we see in Kingdom Hearts Birth by Sleep? It's obviously not Ansem, Seeker of Darkness' artificial Kingdom Hearts or Xemnas' artificial Kingdom Hearts, since those started being created sometime after Kingdom Hearts Birth by Sleep. So, how did this Kingdom Hearts come to be and why was it there during the end of Kingdom Hearts Birth by Sleep?
Xehanort himself answers that question in DDD. He states that he was reckless and rushed his plans in BbS. The χ-Blade in BbS isn't the "real" one, since it has to be made from seven lights and thirteen darkness. That's why the KH shown wasn't a real KH, more like an incomplete version of it.
Why did the time traveling members of the Thirteen Seekers of Darkness go back to their own time? And how are they going to come back?///The Kingdom Hearts Ultimania states: "Due to discarding his body and becoming the Robed Figure, all incarnations of Xehanort gained the ability to travel through time. However, the only ones actually able to do so were the Robed Figure and the one he had given his power to, Young Xehanort." If all incarnations of Xehanort gained the ability to travel through time, then why were the Robed Figure and Young Xehanort the only ones able to do so? That's obviously not the case though since we see Xemnas and Ansem, Seeker of Darkness from the past in present-day Kingdom Hearts Dream Drop Distance. Did they not time travel? Is Young Xehanort gathering them from across time different from them time travelling? If so, then how so? Tetsuya Nomura states that Young Xehanort "borrows" the power to time travel in one interview, states that Young Xehanort "gained the ability" from the Robed Figure by making contact with him in one interview, and the Kingdom Hearts Ultimaniastates he "gave" Young Xehanort some of his own power. Which one is the truth? Did Young Xehanort borrow the power from the Robed Figure, did the Robed Figure give Young Xehanort some of his power, or did Young Xehanort gain the power by making contact with the Robed Figure? Oh, and why did the Robed Figure even need to give Young Xehanort some of his power to travel through time in the first place if all incarnations of Xehanort gained the ability to do so anyway? Tetsuya Nomura states that "Young Xehanort functioned as a “portal”, summoning Xemnas and Ansem each time they appeared. That’s why Young Xehanort was there whenever they appeared." What does Tetsuya Nomura mean by "summon"? I assume he's referring to the fact that a version of you has to exist at your destination in time in order for you to time travel there and that Xemnas and Ansem, Seeker of Darkness would use Young Xehanort as that version of themselves they need to travel to the future. But, like stated above, the Kingdom Hearts Ultimania says that only the Robed Figure and Young Xehanort can travel through time for some reason. And this brings us back to one of the questions I asked above: Is Young Xehanort gathering/summoning them from across time different from them travelling? And if so, then how so?
To be able to time travel you must be only a heart; travel to a time where there is a version of yourself; when you reach the time you wanted to go you can only go foward on the timeline; and you cannot change things that are supposed to happen. The Robed Figure (Ansem SoD) was only a heart at the time and was able to time travel to Young Xehanort period and giving him the same power. With that YX was able to time travel and gather several versions of himself and bringing them all to the end of DDD. When they vanished, everything that has happened will be asleep in their memories and this will aid them on making their decisions through life, leading them to the final showdown in KH3.
In the Kingdom Hearts Dream Drop Distance secret ending, Goofy says that Lea hurried back to Radiant Garden. Why did he hurry back to Radiant Garden?
To begin his traning with Merlin.
It was shown that the "illusion" of Terra we see in Kingdom Hearts 0.2 Birth by Sleep -A fragmentary passage-was there because of Naminé's help. This took place right after Sora went into the pod to restore his memory, so why/how did Kingdom Hearts 0.2 Birth by Sleep -A fragmentary passage- end with how the first *Kingdom Hearts game ended? How does that make any sense timeline wise?
First: the talk between Terra and Naminé is set before the events of 0.2, as stated by Nomura (https://www.khinsider.com/news/SPOILERS-A-special-conversation-before-KINGDOM-HEARTS-0-2-Birth-By-Sleep-revealed-during-First-Breath-concert-6805)
Second: You must remember that time in the the Realm of Darkness is totally different form the Realm of Light. So the meeting between Terra and Aqua happened between the time Sora became a heartless and the end of KH.
How did Terra-Xehanort know about the Chamber of Waking?
Xehanort was trained in the Land of Departure and as a Master, he knows the secrets of the land just like master Eraqus stated to Aqua in BbS. So he knows that castle Oblivion is Land of Departure and about the Chamber of Waking.
How long ago does Kingdom Hearts χ take place? We know the Age of Fairy Tales takes place "more than a hundred years before Sora received the Keyblade", but "more than a hundred years" is a very vague number.
KH χ tells the story of the Keyblade War or the Age of Fairy Tales like you said. But I don't think it makes any difference about when it happened...
How do the Dandelions survive the Keyblade War and the darkness that covers the World after the Keyblade War?
The pc version answers that (Union χ is actually a direct sequence to the pc version) and the american version is about to show us a little bit.
How was present-day Maleficent there during the Age of Fairy Tales during the Kingdom Hearts χ secret ending?
She is searching for a copy of the Book of Prophecies as said on Re: Coded. How she got it and if she has one is yet to be shown.