Here's a plabill for the production of Quo Vadis from which the costumes were filched.
A web site devoted to the acting career of the star of the production gives photos of the costumes here. The web site says: "Joseph Haworth played The New York Theatre in 1900, in the leading male role of 'Vinicius' in Quo Vadis. The spectacular production achieved a long run, and cemented Joe’s reputation as a bankable star of the Broadway theatre."
Here's a publicity photo showing many of the costumes.
The NYPL Digital Library has some stills from the production, including this:
I couldn't find any photos of the car stable, but here is a photo of the old 17th Precinct Station House on 51st Street followed by some photos of horse cars and one of the "New York Theatre."
The author of the Haworth web pages says that the theater "opened as part of an entertainment complex called the Olympia, and marked the birth of a new theatre district in the Long Acre (Times) Square area. Hammerstein’s original idea was a palace of entertainment containing three theatres, a roof garden, billiard rooms, a bowling alley, a Turkish bath, cafes and restaurants. It was a project beset with difficulties, but ultimately two theatres opened in the building: The Lyric and The Music Hall. When Hammerstein sold the Olympia in 1899, The Lyric became the Criterion and The Music Hall became The New York Theatre."