In Britain some children - generally from the upper classes - attend an independent boarding school (known as a "public school") from the age of 11 onwards. In the 1930s almost all such schools were single sex.
The place names are imagined, but are typically English; there was a Newminster Abbey in Northumberland, Condicote is a small village in Gloucestershire. Masefield was born in Ledbury, Herefordshire, and the names suggest a setting somewhere in central England; in 1933 the Masefield family moved to the Cotswolds, which would be a plausible location, but later in the book he says that the golden-coloured Cotswold stone for Abner Brown's house would have had to be imported there.
The Irish terrier is a compact little dog with a rough, red coat. When The Box of Delights was written, terriers had their tails docked soon after they were born, so Barney would have looked much more like the 1915 illustration below.
Before petfood was sold in cans, the figure of the dog's-meat or cat's-meat man was familiar in Britain, pushing his barrow around the streets. There is a picture dating from 1901 in the Museum of London collection. The Cat's-meat man is a character in Hugh Lofting's Dr Doolittle books.
Madam Tussaud's is a wax museum in London, famous for its Chamber of Horrors with depictions of notorious criminals and murderers.