The Uffizi chase scene referred to here is from the 1946 film Paisà, about Allied soldiers during the 1943 invasion of Italy. In the chase scene, a nurse and her friend run through a labyrinthine Uffizi, trying to enter Florence.
The Statue of Liberty scene is from Alfred Hitchcock's 1942 film Saboteur, in which the character Fry flees from the police by climbing out the top of the statue. He eventually falls to his death.
This is a reference to the 1924 film The Thief of Baghdad, in which a thief falls in love with the daughter of a Caliph leader. It starred Douglas Fairbanks, who considered it his favourite work, and he is remembered for his athletic performance in it.
Dick Whittington is a popular character, originating in 1605, that became popular in the 19th century. Though there are many interpretations of the character, the basic story sees Dick, from a poor Gloucester family, move to London to make his fortune, accompanied by his cat. Later, disillusioned and on the verge of returning home, he hears the churchbells ringing from Highgate Hill. Eventually, he does prosper; he marries and becomes Mayor of London.