Shostka is located in present day northern Ukraine. At this time, it is estimated that 1/3 of Europe’s Jewish population lived in Ukraine, which formed a large part of the Pale of Settlement. This area was a target for pogroms, attacks on Jewish communities such as the one in which Fievel and his family lose their home, and these were particularly rife in 1881-84, around the time the film is set. Mousekewitz is a play on the Russian Jewish surname Moskowitz.
Like the mice, expecting the streets to be paved with cheese and their cat oppressors to have no place in America, the expectations of the immigrants were similarly high, and they were similarly crushed on arrival. An Italian saying from the time went, "I came to America because I heard the streets were paved with gold. When I got here, I found out three things: First, the streets weren’t paved with gold; second, they weren’t paved at all: and third, I was expected to pave them.“ The Gilded Age had reinforced the idea of the American Dream as something that was attainable by anyone, with the new money of the industrialists and stories such as those of Horatio Alger being the image of success that was portrayed, while leaving out the harsh reality that many poor workers, including those coming over from Europe to find better lives, faced at the bottom of the pyramid, while the industrialists sat at the top.
Ellis Island is the more famous port for immigration into the USA, but it didn’t open until 1892. When Fievel’s family arrived, they would have gone through Castle Garden, which stands in Battery Park and is now a national monument. The boat in the foreground is owned by the Erie Railroad Company, which would recruit workers almost as they disembarked from the ships that had brought them across the Atlantic. The above screenshot is almost identical to this engraving.
Tanya is given the new name of Tilly when going through immigration, and the myth that names that did not sound "American enough” were changed to something easier for natives to pronounce is a common one. In fact, the officers worked from the ship’s manifest rather than just asking for a name, and they were required to know at least two languages (for more information, see here). It is far more likely that immigrants started to use more American sounding versions of their names themselves, to try to blend in and make it easier for them to integrate.
The Statue of Liberty was indeed being constructed at this time, and was dedicated the next year, 1886. The gold colour of the statue is because it is coated in copper, and was in fact a golden colour when first erected. It is only over time that Lady Liberty has faded to her current green colour, through oxidisation.
The scene where Fievel presses his nose to the window of the schoolroom is based on the experiences of executive producer Steven Spielberg’s grandfather, who told him that “Jews were only able to listen to school lessons through open windows while sitting outside in the snow”.
This Edison phonograph would have used tinfoil inside the cylinder - it wasn’t until 1886 that wax cylinders were patented by Chichester Bell and Charles Sumner Tainter. The march played is The Stars And Stripes Forever, which was not written until 1897, over 10 years after the film is set!
Hester Street, in the Lower East Side of Manhattan, was a centre of Jewish immigrant culture in the late 19th century, and had a 1975 film about the Jewish immigrant experience named after it. Nowadays the area has been absorbed into Chinatown.
Honest John is typical of the Tammany Hall politicians controlling New York politics in the 1800s, most famously Boss Tweed. They relied on immigrants as their main electoral base, and as in the scene above, would go to all sorts of lengths to get votes.
I have always seen the climax of the movie, the mice banding together to construct the Giant Mouse of Minsk, as them knowing that the only way to get rid of the cats is to beat them at their own game - by building their own capital. At this time, the laws protecting immigrant workers were few and far between, and banding together as a community and becoming successful competition to their oppressors was really the only way to combat them.