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Why did they predict a week, half a year, ten years, or however long before the immersion, but it worked in just a few hours? And they spent all ten hours on the flight?

They were asked how long these dreams would take in general, and then there was a list: "a week for the first level, half a year for the second, and ten years for the third." This means that without the ejection, they would have to spend exactly that amount of time on these levels, not that it would take them ten years to implement the plan. Earlier, it was said that "five minutes in the real world equals an hour in the dream," which is why those hours spent on the plane are enough for everything.

Why did the Japanese man age, but the hero didn't, even though they both experienced the last layer?

Because the Japanese man went to the last level after dying on the previous one. And the protagonist did it consciously, using a miracle device that immerses him in the General dream, i.e., he knew he was in the limbo, unlike the Japanese man, who apparently had already lived a life in this limbo before the protagonist went there to find him.

How could the heir lie on the cold floor, dead for ten minutes, and then get up and walk after defibrillation? Because he was found and dropped from a height? Why were they so afraid of "getting stuck forever" on the last layer if the girl and the heir managed to get out of there by jumping?

Yes, because he was found and dropped from a height. They were afraid of getting stuck if they ended up in the limbo after dying; then, they simply wouldn't know that this is not a real world. But they went there consciously, using a device. You can get out of the limbo by using the ejection or by dying, but you can't do it on your own if you get there by dying on the previous level.

Do they necessarily need to hear music, lol? Then why didn't the hero and the Japanese man leave immediately? Because the Japanese man "didn't know"? But he knew before? How did they manage to make the projection of the old man say something meaningful to the heir in the hotel?

Music is needed to synchronize the ejections, in theory. The hero and the Japanese man didn't leave because no one arranged an ejection for them at their level. When the Japanese man ended up in the limbo after his death on the previous level, he didn't know anything anymore and lived a normal life in the limbo until the protagonist found him and reminded him that this world is not real. The "projection of the old man" can, in theory, say anything because it is in the dream of one of the team members.