Do Americans use any other words for the period 1837 to 1901 besides Victorian? Do they also refer to eras in their country as Jacobean, Georgian, Edwardian and so on?

It's somewhat harder for Americans to define their history like that because our "rulers" don't stay in office nearly so long. The part of your era before the American Civil War (up to 1861) we usually refer to as "Antebellum", while a period after the war until about 1876 is called reconstruction here (describing the process of rebuilding the former Confederacy after the war.) After that is a period called by some "The Gilded Age" which lasted up until the end of your Victorian era.

https://en.wikipedia.org/wiki/History_of_the_United_States_%281865-1918%29

Then there's terms like the "Age of Invention" (late 19th-century), "Gold Rush" (mainly on the west coast in the mid-19th century, up to the civil war), the "Homesteaders era" (in the middle of the country, around the same time), and of course the "Space Age" which we're still in.

In addition to what kevinatilusa describes is the fact that America has moved very fast, historically, and many eras are defined in smaller increments. There was the colonial era which quickly became the Revolutionary era, then on the to Civil War(before and after) and Western Expansion, which covers the bulk of the 1800s. The Victorian era wasn't as recognizable here as it was to Europeans, and we have things like the Gay Nineties, the Roaring Twenties, The Great Depression, the War Years, etc. as benchmarks.