Goes the weasel. Her Majesty would not have been amused. Some of the references are now quite opaque, but we can take a fair shot at a few. The City Road had a pawnbroker's shop near its west end and to pop was a well-known phrase at the time for pawning something. So the second verse says that visiting the Eagle causes one's money to vanish, necessitating a trip up the City Road to Uncle to raise some cash.
Nobody is sure.
Some suggest it was a domestic or tailor's flat-iron, a small item easy to carry. Either way, it seems to have been a punning reinterpretation of the catch line from the older dance. The third introduces the monkey, one sense of that word being a nineteenth-century term for a drinking vessel in a public house, which makes sense in context. It could be one belonging to an organ-grinder, an itinerant musician who played a small portable organ, of whom there were many at this period. But I suspect there are topical or slang references in there that are now lost.