AP Paul Morigi photo of Kevin Gover 'Americans' exhibition at Smithsonian National Museum of the American Indian Photograph: Paul Morigi

"Indians are less than 1% of the population. Yet images and names of Indians are everywhere. How is it that Indians can be so present and so absent in American life?"

This is the question posed by Americans, a new exhibition at the Smithsonian National Museum of the American Indian in Washington, exploring how Native Americans have been central to America's sense of itself even as they were systematically persecuted, marginalised and erased.

The myth-busting show contains an array of nearly 300 objects and images of Indians and Indian stereotypes. They include a Tomahawk flight-test missile, a 1948 Indian Chief motorcycle, a Washington Redskins football team baby blanket, photos of presidents and celebrities wearing feather headdresses, footage from westerns and scale models of Chinook, Kiowa and Apache Longbow helicopters.

The myth-busting Americans show contains an array of nearly 300 objects and images of Indians and Indian stereotypes. Photograph: Smithsonian

Americans was conceived during Barack Obama's presidency but arrives in the shadow of Donald Trump. Many Native American activists praised Obama for doing more than any other US president to recognise their grievances, including the government's historical neglect of treaty obligations. Trump's biggest impression so far, as a bang-up-to-date digital display acknowledges, is using the term "Pocahontas" to insult Senator Elizabeth Warren over her claims to Cherokee ancestry.

But even as Native Americans find their rights under renewed threat, for example from the Keystone XL and Dakota Access oil pipelines, their cultural voice is growing stronger. Last month the Cleveland Indians baseball team announced that they will drop the red-faced Chief Wahoo caricature from their uniforms next year, bowing to decades of complaints. One of Washington's leading theatres is staging Sovereignty, a new play that incorporates Cherokee language and is written by Mary Kathryn Nagle, a playwright, lawyer and citizen of Cherokee nation.

Speaking on a panel with the author and cast on the first day of rehearsal, Molly Smith, artistic director of Arena Stage, said: "I have to tell you there's a big upswing in Native American plays being produced around the country. It is your time. So it's pretty thrilling that this voice is now being heard."

She added: "I think it's a big part of our national shame that a lot of these stories have not only not been uncovered but that there has never been a real apology to Native Americans here in our country for what happened." Until the stories are uncovered, she added, there will be no change.

The play darts back and forth between the 1830s and near future with characters including President Andrew Jackson, who signed the Indian Removal Act and whose portrait now hangs in the Oval Office, and a violent drunkard wearing a "Trump" T-shirt.

Kyla Garcia, who plays a character based on Nagle, recalled an incident in which Trump branded Warren "Pocahontas" during a White House ceremony honouring Navajo code talkers, who served in the marines during the second world war and used the Navajo language to transmit encoded information.

"It just inspired me with fury," Garcia said during the panel discussion. "Because it solidified even more why we're telling this story and why we're up here. Because for that to be happening and for people not to know the history behind that: there were some people who didn't even realise Andrew Jackson was in the back in that moment, and what that means.

"So to be at the forefront of these kinds of stories right now I feel is the most sincere form of activism I can do and that makes it an absolute honour and privilege to be telling this story, and to be educating people as I go. When people say, 'What are you working on?', I'm like: 'You have a minute?'"

The backdrop to Sovereignty is Jackson's refusal to enforce an 1832 supreme court decision to uphold the right of Cherokee nation to prosecute anyone who came on Cherokee lands and committed a crime. In 1978, the supreme court formally took that right away, ruling that tribal nations historically did not exercise criminal jurisdiction over non-Indians. Aware that tribal governments were now powerless to prosecute them, non-Indian offenders acted with impunity and violence against Native women soared.

The Smithsonian exhibition. Photograph: Supplied

Nagle writes in the programme notes: "Today Native women face rates of domestic violence and sexual assault higher than any other population in the United States. It took 140 years to full come into effect, but Andrew Jackson's campaign to eliminate tribal jurisdiction has reaped devastating, life-and-death consequences for Native women."

In an interview on Friday, Garcia, who has indigenous roots on her father's side, added: "The experience of Native American women has been missing from the #MeToo movement when they suffer the highest rates of abuse. The only way to understand an epidemic is to acknowledge those who are most vulnerable."

Trump, she continued, is far from alone in his fundamental ignorance of Indians' place in the American story. "The leader in office represents the majority of the American public. For him to make jokes about Pocahontas, all that tells me is he doesn't understand history."

Pocahontas has been portrayed and warped by popular culture, from a frieze in the rotunda of the US Capitol to Disney's 1995 animated film, but the reality is less romantic, the Smithsonian display shows.

She was born in 1595 in what is now Virginia. Her father, Powhatan, was the leader of a powerful confederacy when, in 1607, a hundred Englishmen landed. The child Pocahontas impressed the English with her self-confidence and, in her teens, she brought food to save the settlers from starvation.

But Pocahontas was abducted by the colonists, learned English, converted to Christianity and married the tobacco planter John Rolfe. She travelled with him to London, where she was presented to society as a "civilised savage" and restored the confidence of English investors in the new colonies. She died young, aged only about 21.

Such was the cultural mythology around her that when Virginia passed the Racial Integrity Act of 1924, criminalising interracial marriage and requiring that every birth be recorded by race ("white" or "coloured"), it made exceptions for whites who proudly claimed Native ancestry from Pocahontas. In a startling example of humans' capacity for hypocrisy, Native Americans themselves did not enjoy this privilege.

The museum also investigates the Indian Removal Act of 1830, which proposed that Indians living inside the country's then boundaries should leave in return for payment and new land west of the Mississippi river. This was despite the US having made treaties with many Native nations in which it recognised their sovereign territories. One consequence was the forced relocation of about 16,000 Cherokees in 1838–39, which became known as the Trail of Tears.

Though Jackson signed the legislation with relish, he also reflected a school of thought that went all the way back to Thomas Jefferson, who referred in the Declaration of Independence to "merciless Indian savages" and has been dubbed "the founding father of Indian removal". Overall, the vast, systematic campaign of forced displacement affected 67,000 Indians across the terms of nine presidents at a cost of $100m.

Cécile Ganteaume, associate curator at the museum, said: "It shows that Andrew Jackson wasn't alone in his thinking. That's why we wanted to contextualise the Indian Removal Act, which he signed. There were others and the country went into this with eyes wide open."

The exhibition raises the question of why US popular culture regards Native Americans as authentically American yet simultaneously objectifies and exoticises them. Their image became fodder for advertising and supermarket products in the post-war consumer boom. They are everywhere and nowhere.

"Most Americans think they have nothing to do with American Indians and American Indians have nothing to do with them," said Ganteaume. "We're saying the exact opposite is true: they have a deeply entangled history.

"No other country in the world, as far as we know, is so fixated with with one segment of the population that it is constantly creating representations of them. It's a deep paradox: for Americans, American Indians are essential to their own sense of themselves, but while imagery of American Indians is everywhere, it's a curtain to prevent Americans knowing who American Indians truly are."