Short answers, pulled from the story.
The Prince of Wales Theatre first opened in January 1884, when architect C.J. Phipps built the Prince's Theatre for actor-manager Edgar Bruce. It was renamed the Prince of Wales Theatre in 1886 after the future Edward VII.
The Beatles performed at the Prince of Wales Theatre on the 4th of November 1963, during the Royal Variety Show. They performed in the presence of the Queen Mother and her younger daughter.
The theatre has been rebuilt twice. The original 1884 building was demolished and replaced with a new Art Deco theatre designed by Robert Cromie, which opened on the 27th of October 1937. The building was then extensively refurbished in 2004 by Sir Cameron Mackintosh.
The Book of Mormon is the longest-running musical at the Prince of Wales Theatre, having celebrated its 4,000th performance there on the 6th of June 2024. It first opened at the theatre on the 25th of February 2013.
Henry Savile Clarke and Walter Slaughter's musical Alice in Wonderland, considered the first major stage production of Lewis Carroll's books, debuted at the Prince of Wales Theatre on the 23rd of December 1886. Phoebe Carlo played the title role, and Carroll himself attended a performance seven days later.
Sir Cameron Mackintosh owns the Prince of Wales Theatre. He oversaw an extensive refurbishment of the building completed in 2004, which raised seating capacity to 1,160 and fully rebuilt the auditorium.