[Note 329.7] "He'd near drained Fortunes cup":- may be a reference to the practice of reading fortunes by interpreting the pattern of leaves left in the cup after the tea had been consumed. Reading the tea leaves became popular in the 19th century (ref
http://tarotamber75.wordpress.com/2013/01/19/brief-history-of-tea-leaf-reading/) It was also the kind of superstitious practice likely to draw the sneers of the upper classes.
[Note 329.8] "But then he did not see / Though once a footman, any fun / A running one to be" - a running footman was employed to run for some specific purpose, as to deliver messages, be in attendance upon a coach or carriage, etc. [OED]
[Note 036.0] The British Library record suggests that this was one of a collection of songs called "Bonny Boys" printed around 1850.
The Roud Broadside Index includes 23 entries for Bob the Groom which suggests it was a popular song; an idea supported by the existence of the broadside John the Coachman, A sequel to Bob the Groom (Roud V10197)
[Note 329.9] "For rouges to bring all things so round As James did in the Square" :- The mixing of the social classes that was brought about by the railways caused great unease. The footman's eventual fall back to his original lowly situation would have gratified the upper class audience at whom this song was clearly aimed.