1. Yes, Sora and Riku actually traveled through time. As Young Xehanort and Master Yen Sid explained, in order to travel through time you must leave your body behind and arrive at a point in time where a version of you already exists. Yen Sid put Sora and Riku's bodies asleep so he could send their hearts back to the point in time right before Destiny Islands was destroyed by the Heartless, which allowed them to experience everything in their younger bodies. Once Destiny Islands fell to Darkness like it did in KH1, Sora and Riku were able to gain access to the first sleeping keyhole in the past, which allowed them access to the Realm of Sleep (and it probably doesn't matter at what point in time you enter the realm of sleep, most sci-fi and fantasy agree that dreams can transcend space and time, so as far as everyone in the present was concerned, Sora and Riku's adventure was happening in the present as soon as they entered the realm of sleep).
2. Traverse Town may have never been destroyed by the Heartless, but it is a special world of it's own with special properties. Like The End of The World, Traverse Town is made up of the remnants of various worlds that were destroyed by the darkness. This allowed it to act as a safe haven for anyone who survived the destruction of their world and corridors of darkness would regularly open up and close allowing those individuals to find their way there. This is how Sora found his way to Traverse Town in KH1, before he got swallowed up by the dark storm he fell into a corridor of darkness and ended up in Traverse Town. Not only that, but Traverse Town is one of the few worlds that exist in The Realm Between, between the Realms of Darkness and Light, allowing it to be easily accessible to denizens of the Light before they are captured by the Darkness. Because of all of these traits, Traverse Town can be described as a special world that appears to those who need it, a lamp post on a dark road so to speak (kind of explains the whole nighttime scene I suppose). It's because of this that it appears in 3D. As Joshua explained, something happened in their world (Shibuya) that threatened the existences of his four friends who were forced to play the Reaper's Game (I'm going to assume that seeing how they're all chummy with each other now, the events of the TWEWY already happened and the Reapers went rogue and forced them into the Underground again...which probably meant murdering them...again...and possibly made the game so impossible and unreasonable that they couldn't complete it...or the Heartless were invading Shibuya at some point and without that world and it's alternate planes of existence the crew couldn't survive, complete the game, or something like that. Point being, some kind of crisis happened in that world and I can only assume that it was either caused by the Reapers or the Heartless...or both.). So being the deity that he is he managed to save what little he could of them and wounded up in the Realm of Sleep. In order to give them a plane to exist on, he used the dreams of Rhyme to try and create a world that they could survive in, a world that they needed. So naturally it took the shape of Traverse Town, and since Joshua managed to save the dreams of Neku and the others, they were able to take shape and form and were restored within this dream Traverse Town. However, interestingly enough it became split into two, likely because it was formed when Sora and Riku first made the dive into the Realm of Sleep. Since during the whole adventure Riku was in Sora's dreams while Sora went through the actual Sleeping Worlds, it's likely that the sudden creation of two dreamscapes caused dimensional hiccup (or a tactical boo-boo, whichever babble you think sounds funnier), so when the dream Traverse Town was created, it got stuck between the Realm of Sleep and Sora's dreams (which also makes sense since he and Rhyme are both pretty big dreamers). So when this happened the team got split up, Neku and Rhyme in the Realm of Sleep and Joshua, Shiki, and Beat in Sora's dream. Once Joshua was able to use Rhyme (whose dreams as I mentioned were the source of this Traverse Town) as a Portal between the two splintered versions of the town, he was able to take Rhyme and Shiki between them so they could be reunited with their partners. Why he couldn't bring them all into the same version is likely because it would have been too difficult even for him, since he wasn't even in complete control over how the town was created, let alone split apart. But with everyone reunited with their partners they could at least have a chance at completing the game, which the Reapers are apparently still in charge of and don't seem to be letting up on even against Joshua's wishes. After Sora and Riku finished awakening all the Sleeping Worlds, the fate of the Shibuya team is unknown, since we don't know if they have a home to go back to or not. And even if they do, we don't know if it would be safe for them to ever return, since they're only still here because the Realm of Sleep gave them shape. That's probably something Joshua will have to ponder over for a while longer. Also, it's never clear whether this would be considered a pseudo Traverse Town made from dreams or just a dream extension of Traverse Town that can appear for those who are lost in the Realm of Sleep from the waking world and are not part of any other Sleeping World. In either case, it appeared because it was needed, much like it does in the real world.
3. As I said before, in order to gain easy access to the Realm of Sleep, Sora and Riku had to be sent back in time to when a world they had been on was being destroyed by the Darkness. And in order to time travel to the past, you must leave your body behind in the present and have a past version of you at your destination so that you may exist as your past self for that adventure. Because the Destiny Islands were destroyed in the past, Sora and Riku were present when it happened, and Destiny Islands fell to slumber until it was restored by Sora, the circumstances were favorable enough to act as a gate for Sora and Riku to enter the Realm of Sleep (the only other alternative would be to cast a world into darkness in the present and...yeah, kind of not their thing). So when Yen Sid sent them to the past on Destiny Islands before the storm hit, Sora and Riku awoke in their younger bodies (and I guess Riku wanted to cut his hair in his past body too because...reasons). So in a sense it is due to Yen Sid's magic that they appear younger, but that's because they actually followed the rules of time travel and they are actually in their younger bodies. It's possible that maybe Yen Sid did give them new adventuring clothes to better harness their dream powers (since he and the good fairies do have clothing magic), but it's also likely that their clothes changed entirely due to the outside forces at work (Sora and the Recusant's Sigil, the sigil likely coming with the outfit, and Riku and the Spirit sigil, the sigil representing Riku's state as a Spirit Dream Eater and the clothes likely came with it as well). Point being, the bodies they have in 3D are the same bodies they had in KH1, their hearts and minds however are from the present.
4. Yes, Riku was in Sora's dreams from the moment they first Dived all the way to when he appeared in front of Memory's Skyscraper in the real world. As Ansem SOD explained, when Riku spotted the "Man in the Robe" (which was Ansem as just a heart), he realized that something was not right, so when he and Sora opened the first Sleeping Keyhole that lead to the Realm of Sleep, he unconsciously used the Keyhole to enter Sora's dreams instead so that he may be able to protect Sora, causing him to become a pseudo Spirit Dream Eater. That means that while Sora had been dreaming through the Sleeping Worlds all this time, the whole time Riku was dreaming through Sora's dream of the Sleeping Worlds. It wasn't until Riku defeated Ansem and Sora fell into a dark slumber that Riku was finally able to escape both the Realm of Sleep and Sora's dream. It should be noted that while Sora was in the real World That Never Was, he was forced into sleep again, placed into a nightmare version of the world created by the Organization (essentially they just gave him a nightmare), so the World That Never Was that Riku was in was the dream of Sora's nightmare of The World That Never Was, which was how he was able to access it. Once Sora had fallen into slumber, Riku's connection to him was cut off, allowing him to appear in the outside world, and since they were no longer in the Realm of Sleep, he appeared in the real world where Sora was. It's also interesting to note that while Sora is slipping further into darkness the world that Riku is in gets more and more dark and distorted during his fight with Ansem until he is finally back in the real World That Never Was.
5. The first Keyhole that appeared in Traverse Town was used to "awaken" the town and unlock the rest of the way to the other Sleeping Worlds. However since the town is still needed for Neku and the others it still remains. So when Sora and Riku use the Keyhole again on their second adventure there, it's not used to awaken the town but rather to unlock the path to the next two Sleeping Worlds, meaning that while they have technically used "eight" Sleeping Keyholes, they really only "unlocked" seven. This leads to further confusion as to what the fate of that world and the Shibuya team currently is.
6. Sora and Riku really were on the Destiny Islands, but it was back in the past, the very day that it fell to darkness. Yen Sid used his magic to take Sora and Riku's hearts and consciousnesses into their past bodies so that they would be able to dive into the Realm of Sleep when the Destiny Islands became a Sleeping World. The only "illusion" of the world was the Phantom Ursula that showed up to battle Sora and Riku, created by Yen Sid as their first trial of the Mark of Mastery Exam. It disappeared once it was defeated and the world fell to darkness.
7. The "Sora" that Riku finds in the dream of a dream version of The World That Never Was isn't the "real" Sora, it's more of a existential metaphorical representation of Sora's subconscious as he's experiencing the nightmare, and it also possibly represents the "nexus" of his dream of the Sleeping Worlds that Riku has been traveling through, meaning that that "Sora" represents the center of Sora's dreams and Riku has finally caught up to his subconscious mind (which is likely why Sora was able to faintly hear Riku's voice during his nightmare). So the Anti-Black Coat Nightmare that Riku fought also acted as a physical manifestation of the nightmare that Sora was having, which makes sense since a majority of the people he comes across are in the Organization's coat, the nightmare was brought on by Xigbar and Young Xehanort, he's fighting Xemnas, and the Nightmare actually mirrors some of Xemnas and Young Xehanort's attacks. The Nightmare emblem appears on the back to also mirror Riku in a way, as while Riku is represented as a Spirit to protect Sora from Nightmares that would harm him, the Anti-Black Coat Nightmare is the strongest one that has ever been encountered by the two and it exists to drag Sora further into the darkness of his dreams and keep Sora trapped in a dark slumber that he can never wake up from. Riku was able to defeat the Nightmare, thus leading to Sora's actual nightmare to end, but once Sora began to fight Xemans to the point of exhaustion he was already starting to fall into the darkness, which started to cut Riku's connection to Sora's dreams, which is what caused the Sora image to disappear and the dream Riku was in to distort and become more dark and twisted while he fought Ansem, who probably also had a hand in the nightmare.
I hope this helps clear things up for you. I too also love Kingdom Hearts for it's complexities, it's fun to have a series that acts as a puzzle rather than one that spells everything out for you the first time. It makes the series more deep and fun to explore. Also trying to explain things sort of helps you better understand it yourself, so I feel that I've learned something from this too!