The book in question is The Twenty-one Balloons. Excerpts below are from the Wikipedia article and are cited in accordance to its license.
I recall the Jules Verne story being mentioned in this work, but as fact not fiction. Something like "I realized I could beat the record of 80 days to get around the world".
The introduction compares two types of journeys: one that aims to reach a place within the shortest time, and another that begins without regard to speed and without a destination in mind. Balloon travel is said to be ideal for the second kind.
The reference to Jules Verne's story is actually kind of after the fact. The Professor makes his trip and afterwards, the balloon society realize that he made his trip in less than 80 days, so they celebrate that.
He lands on Krakatoa, which has a diamond mine and a Utopian society where people invent their own house according to their wishes. One of them does have the sinking furniture although the Wikipedia article does not mention it.
Each family has been assigned one of the first twenty letters of the alphabet, and lives in its own whimsical and elaborate house that also serves as a restaurant. The Krakatoa society follows a calendar with twenty-day months. On "A" Day of each month, everyone eats in Mr. and Mrs. A's American restaurant; on "B" Day, in Mr. and Mrs. B's British chop house; on "C" Day, in Mr. and Mrs. C's Chinese restaurant; on "D" Day, in Mr. and Mrs. D's Dutch restaurant, and so forth. Sherman's first friend on the island, Mr. F, runs a French restaurant containing a replica of the Hall of Mirrors. The houses are full of incredible items, such as Mr. M's Moroccan house, which has a living room with mobile furniture that operate like bumper cars. The children of the island invented their own form of amusement that combines elements from merry-go-rounds and balloon travel.
One of the funny bits of the book is that the Professor is essentially told that he can't join them because it would be too confusing to have someone say "Hey, U" and have everyone turn.
It may also have been that he was shipwrecked on the island and left in a balloon, traveling fast when the volcano erupts.
When the volcano on Krakatoa erupts (in its famous real-life 1883 eruption), the families and Sherman escape on a platform held aloft by twenty balloons (The book's title refers to these balloons in addition to Sherman's original balloon). As the platform drifts westward around the world, the families parachute off to India and Belgium to start their new lives. Sherman remains on the platform and finally descends onto the waters of the Atlantic Ocean, where he is rescued.
I think I've covered your main points. It's an excellent book, very funny and with a decent amount of scientific thought to it (although the bit about the diamond axe doesn't make much sense. Diamonds shatter far too easily) and some very nice illustrations.