1. The "American" accent is a combination of several factors, and there were a lot of regional differences. As immigrants settled, they combined their original (term used loosely) accents, their neighbors' accents, and then something that linguists call "hyperpronunciation"- which is when children overemphasize a certain sound not necessarily emphasized in their parents' speech as a way of learning it- which can result in children having a slightly different "accent" that accumulates over time without correction, or under influences from peers. For example, in western Virginia there used to be speech patterns that combined 18th c English, Scots (which is not the same as Scots Gaelic), Irish, certain German dialects, Welsh, and several Native languages. They also held onto archaic words and pronunciations due to isolation that hadn't been heard anywhere else in 200 years. These have largely disappeared (thank you, radio and television), but people did some interesting studies during the New Deal era on Appalachian language and folkways if you're curious.

Specifically- a lot of historians point to the 1775 dictionary published by John Walker to say that British English changed and American English didn't, and then common Americans just added new words. Specifically, the "r" disappeared in British speech in most places and was added back in others- listen to Major Winchester's accent on MASH for what an 18th c English person might have sounded like, very different to modern British English. Hence one reason Southern Americans may not pronounce r's- Southerners retained connections to England long after other parts of the country and tended to be more ethnically homogenous with fewer language influences than say, New York City.

Regarding Rangers- not in the 18th c. The "men who could not be stampeded" were basically a product of early settlement of Texas in the 1820s-1840s. They were originally organized by Stephen Austin and Moses Morrison to defend coastal settlements from Indian raids, but they weren't officially called Texas Rangers until after the Civil War. The more common term was the Texas Militia, neatly combining an American concept of home defense (see Rev War) and a Spanish military organization of partisan Rangers. They became known and reviled along the lines of the Scottish Black Watch- you never knew who was in charge or whose side they were on, if any.

Texas settlement started in the 16th c, but Americans figured out it was fertile cotton land in the 1810s and 1820s shortly after the Louisiana Purchase opened up the West. It was an odd arrangement because Spain traded Louisiana to France and France promptly sold it to the US- without most of Louisiana realizing they weren't Spanish anymore, much less American. Then a lot of American settlers got the Manifest Destiny (1839-1845, attributed to an Irish journalist named John O'Sullivan) bit between their teeth and spread out up the fertile river valleys from the coast. This was complicated by the Adams-Onis Treaty of 1819 in which the US got Florida in exchange for Texas, but Texans kind of liked their new lands and refused to leave or be governed as Spanish citizens. Spain and Mexico broke up in 1821 when Mexico declared independence, but Mexico had been generally autonomous since at least the turn of the century. Then, in 1830, Mexico abolished slavery- which annoyed Texas planters with slaves, who then stared their own independence movement.

American politics at the time are also complicated- the Jackson presidency could be its own soap opera.

So, I will preface this by saying I haven't seen The Revenant (my area of expertise is basically 1820-1860, so I have a tough time with movies about that period), so I can't speak to any specifics of that film. However...

1. /u/histpresarchitect did a good job with this one

2. Not as far as I know. The Republic of Texas was founded in 1836, and the Mexican War of Independence ended around 1821, so it seems pretty unlikely. Perhaps someone with an expertise in military history, law enforcement, or Texas can shed further light.

3. "What was going on in the US during the 1810s-1830s?"

This is, as you know, a very broad question. You will know that the United States fought a war with Britain, known here as the War of 1812, from 1812-1815. This war was over several questions, but historians today tend to see it as mostly a kind of second Revolution - it settled the question of American independence once and for all. Americans did not realize their goals of conquering Canada, while the British were forever shut out of the Eastern half of the United States. The border between the U.S. and Canada all the way to the Pacific would later be settled by treaty in 1846, though there remains boundary dispute in the area of Maine. So, one thing that was going on in this period was the territorial borders of the United States became legitimized and further codified.

Speaking of which, the Louisiana Purchase of 1803 had greatly expanded the country's territory, so settlers were streaming to fill that land, often at the expense of local American Indian populations. One of the most infamous results of this was the Indian Removal Act of 1830, which removed the Cherokee and other Southern Indians from their ancestral lands to distant land west of the Mississippi. All of this white settlement of the Louisiana Purchase area led to a need for a way to ship agricultural goods to ports in the East. In particular, the rich agricultural lands of the Great Lakes region had access to neither the Mississippi River (too far West) or any sort of navigable rivers that headed East, so the country decided to build the Erie Canal, one of many internal improvements that sought to establish a viable system of canals, turnpikes, and eventually railroads to economically integrate the country more broadly.

Politically, this is the time known as The Era of Good Feelings, in which the collapse of the Federalist Party led to a brief period of one-party rule under the Democratic-Republicans. This led to the era of Jacksonian Democracy, when Andrew Jackson, the hero of the War of 1812, avowed Indian hater, and first president from outside the original 13 colonies was elected. Jackson and his followers strove for greater democratic participation for common people (well, white men anyway), leading to an era of boisterous civic engagement that saw many of the social and political norms of the 18th century fall away. His opposition, consisting of elements of Jefferson's Democratic-Republicans who shunned mass democracy and the remnants of the old Federalist Party, coalesced into the Whig Party, which would be one of the two major political parties until it collapsed and was replaced by the Republicans in the 1850s. Jackson favored elected judges, decried a central bank, and favored territorial expansion. This is also an era of the rise of the political machine, institutions which would come to dominate politics, particularly in cities, for the rest of the nineteenth century.

Socially, this is the first great era of urbanization in American history. As the farms of the Northeast could no longer compete with new Midwestern producers, which had better soils and flatter land, the people living on those farms either joined the movement westward or began to move to the cities, mostly to work in the new mercantile houses (and the very nascent manufacturing) fueled by the agricultural boom and Erie Canal. This led to the development of some of the first mass culture in the country, centered especially around theatre. The minstrel show, for example, has its origins in a newly mobile urban culture of the 1830s. In the South, the cotton gin opened vast new areas to cultivation, greatly increasing the depth and breadth of slavery. Southerners hoped to expand their institution further West, consolidating political advantage of slave states in Congress. Northerners hoped to resist this. Thus, this is also an era of increasing sectionalism The nullification crisis of 1832-3, in which South Carolina sought to invalidate a federal law, also sees the first stirrings of rebellion on the horizon.

Culturally, the 1820s and 1830s saw the rise of some of the first American intellectuals and authors who would achieve international recognition. Most notable among these were Ralph Waldo Emerson, Henry David Thoreau, and the Transcendentalists, a literary and philosophical movement that translated many of the precepts of European Romanticism, especially its German variant as represented by Goethe. Other American authors with international renown include James Fenimore Cooper and Washington Irving, though Emerson was by far the best-regarded American literary mind of his day. In American art, this is the era of Romanticism, followed by Thomas Cole and the Hudson River School. The HRS sought to represent the new landscapes opened up by Manifest Destiny, and has often been seen as a way of achieving cultural nationalism during a fractious time.

I hope that is a decent overview. I am not very good with foreign affairs, but in general things were quiet after the War of 1812 and before the War with Mexico. I am sure someone can fill in the general treaties and whatnot if you want. I can also try to answer other questions, in particular about social and cultural history, my own specialties.

Sources:

General:

Howe, Daniel Walker. What Hath God Wrought: The Transformation of America, 1815-1848

Wood, Gordon S. Early National America, 1789-1815

A couple of great sources on the cultural phenomena I mentioned:

Lott, Eric. Love and Theft

Miller, Angela. Empire of the Eye

Howe's book is magisterial, one of the finest works of synthetic history I have ever read. If you are interested in this time period, you should read it first and foremost. He will also point you in the proper direction for sources on any number of topics relevant to the period.