ZUN said about Mugenkan that "it was genuinely with the background in the bedroom".
The mansion itself apparently lies on the boundary between Gensokyo and the Fantasy World, though whether this world is the same as the World of Fantasies from Story of Eastern Wonderland is unknown. Physically, this boundary is located beneath the Lake of Blood somehow, which is in the middle of a lake on a mountain behind the Hakurei Shrine.
The fifth stage was in the deepest part of Mugenkan, but it is unclear whether or not the location of this stage is set inside Mugenkan, since manual says its location is "???" and the background is all black with some star-like glows. The stage theme is "Dream Land" and the unused version is called "Border Land", so it is possible that it is in the Fantasy World, or in another Fantasy World. Incidentally, it seems as if a human stays at where stage 6 is for a few hours, they will turn into a mist of atoms.
At the end of the third stage of Lotus Land Story, there are lots of blue tiles on the ground in front of the door of Mugenkan, and the gatekeeper of the entrance Elly throws some of those tiles at the beginning of the boss battle.
On the fourth stage, there red checkerboard pattern floor that continues down a corridor, and somewhere in the middle of the stage, there are two six-starred magic circles.
On fifth stage, there are three floating blue checkerboard platforms, and below there are big blue transparent stars that moves down the screen constantly. Also below the stars, there appears to be a number of watches.
During stage 4 of Lotus Land Story, the player first fights either Reimu Hakurei or Marisa Kirisame, depending on which character the player is using. Following that, they battle Yuuka twice, the second time in some other sort of Fantasy World.