There was no anagram.

The most significant anagram of "Albus Percival Wulfric Brian Dumbledore" is "Albus Percival Wulfric Brian Dumbledore". The full name of Albus Dumbledore was first revealed in Harry Potter and the Philosopher's Stone, long before these supposed theories were posted. No anagram has ever been found for said name, and the name seems quite long for an anagram that anyone could be expected to predict. It would be quite odd for J.K. Rowling to show an anagrammed name in the first of seven books and only ever mention said anagram in a chat with fans who don't know they're speaking to the author.

J.K. Rowling likely wouldn't want to use an anagram as a plot device again after "Tom Marvolo Riddle", lest she is being accused of being lazy and unable to come up with compelling ways to hide a villain's identity. In addition, Tom's name included the fake word "Marvolo", whereas all of Dumbledore's names actually mean something in real life or mythology. For example, Rowling has shared not only the origin of but also why she chose "Dumbledore" as Albus's last name.

Other names mean something -- Dumbledore, which means "bumblebee" in Old English...seemed to suit the headmaster, because one of his passions is music and I imagined him walking around humming to himself.
Barnes and Noble interview, March 19, 1999

There has also never been any hint to Dumbledore's name being an anagram in the books. While "Tom Marvolo Riddle" was never specifically foreshadowed as an anagram either, the name "Riddle" could've clued some readers in as to Tom's name having a deeper meaning. On the other hand, Albus Dumbledore has never been revealed to be an anagram of any kind even in Deathly Hallows, the final book.

Most of her theories were bogus.

While there is evidence that the person we see in the chat logs is not actually Rowling (Dacium's comment from 15 years ago sums it up well), it's clear either way that the "theories" she posted were incorrect. She continually stated that the ninth chapter of Goblet of Fire, a chapter primarily relevant to the plot of the book on its own and the events leading up to the return of Voldemort, was significant to the seventh book, even claiming that it had to be rewritten 1213 times. She even went so far as to state that Remus Lupin is the "HALF BLOOD PRINCE", which both relates to the sixth book rather than the seventh (while Rowling said that she posted theories about the seventh book during her chat) and is obviously incorrect to anyone who has read Half-Blood Prince. If any of what she posted was ever true, it was scrapped before the last two books were released.

Not as far as we know.

No middle names have ever been revealed for either Aberforth or Ariana (although who knows, if someone asks JK Rowling on Twitter, she might make some up for you).

According to his Pottermore fact file - though I'm not sure if this particular part of Pottermore is counted as canon - Aberforth's full name is Aberforth Dumbledore, so he has no middle names. I'm not aware of any information at any level of canonicity on Ariana's middle name(s) or lack thereof.

As for why Albus had so many names: again, there's no canon statement to address this, but I can think of a few possible explanations.

• He was the oldest. When he was born, his parents might have been over the moon and given him lots of names, not knowing whether they'd ever have another baby to name. By the time his brother and sister were born, maybe they'd decided that such long-winded names were a bit silly. Note that "Percival" is named after his father, a common practice among parents at the time.
• He gave himself all those middle names. It's possible in Muggle society to give yourself a middle name that you didn't have as a child, so why not in wizard society too? It would fit Albus's character - "Nitwit! Blubber! Oddment! Tweak!" - to tack a few extra names onto his own, and it also fits with the relative grandeur of his career. Aberforth, content to remain in the shadows of the Hog's Head, doesn't seem like the type to bother giving himself extra names; and of course Ariana didn't survive into adulthood.