There are several elements of this question that I will attempt to tackle individually. First, the quickest one: pirates. The romantic image of pirates came into existence before the so-called "Golden Age of Piracy" even ended in the early 18th century, informed by the works of Daniel Dafoe and expanded on over the centuries. Here is a list of discussions of the public image of pirates.

Next, let's turn to Mongols. Simply put, Mongols have not been as significant a part of the cultural zeitgeist in the English-speaking world. Depictions of them or other later steppe nomads are not as common and primary research translated into European languages is relatively sparse. Early modern works like Christopher Marlowe's Tamburlaine depict the Turko-Mongol ruler Timur as a bloodthirsty and ambitious conqueror, but also one to be respected for his defeat of the Ottoman Turks. Later references to Mongols and their empire come in Samuel Taylor Coleridge's opium-fuelled poem Kublai Khan and John Wayne's disastrous portrayal of the character in the 1956 film The Conqueror. The movie's yellow-face Western style story was not well received, and it remains one of the biggest flops in movie history. Additionally, because the movie was filmed downwind of a nuclear testing site, the production was linked to the cancer deaths of 91 members of the cast and crew.

However, the portrayal of Mongols and steppe empires outside of the West is often very different. In Mongolia, Chinggis Khaan is regarded as a national hero and is honored by, among other things, a massive statue near the Mongolian capital of Ulaanbaatar. In recent years, portrayals of the Mongols have more sympathetic, or even more grounded in history. Many of these are more recent than the 20 year rule on this sub, but they include video games like the Age of Empires, Civilization, and Mount and Blade, movies like Mongol (2007), or music like The Hu and Tengger Cavalry.

As for Vikings, they are on their second round of cultural rehabilitation. The first major fascination in Europe with Vikings happened in the Victorian era. Starting in the 1830s, British and later German writers, composers, and artists took the growing body of scholarly work on pre-Christian Scandinavia and turned into something of a pop creation myth for the greatness of their respective cultures. Particularly, historians like Andrew Wawn see the translation of Frithiof's Saga in the late 1830s as a significant point in the transformation of the pre-Christian Norse from rapacious pagan villains to exemplars of Anglo-German manhood. Viking sagas became a staple of school-boy texts in the late 19th century, and the subject of countless painters and musical works, perhaps most famously Wagner's Der Ring des Nibelungen (from which Ride of the Valkyries is immediately recognizable). I mention Wagner in particular because his work relates to the subsequent fall from grace of Vikings in the mid-20th century, as their imagery was co-opted heavily by Hitler and the Third Reich. After the war, serious portrayals of vikings suffered from a taint of Nazism, but in the later half of the 20th century, whimsical, lighthearted vikings like Hagar the Horrible, the Minnesota Vikings football team, and Terry Jones' Erik the Viking softened the nationalistic image and brought it back into the acceptable mainstream. (There is a great article on this, but I am having a hard time finding it. It was published in 2011 along with the release of Skyrim.)

Finally, we come to conquistadors. First, it should be noted that, unlike the other people discussed in this question, the conquistador violence from the 16th century is still very evident in the societies it most affected. Indigenous people in the Americas still suffer continual violence as the result of European colonialism. As such, positive portrayals of conquistadors are likely going to be met with more backlash than the others in this post.

However, it is always worth noting that, in English-speaking countries, the status of conquistadors has also been heavily informed by the "Black Legend." The Black Legend refers to the way mostly British writers have characterized the Spanish conquest of the Americas as particularly cruel and brutal, as compared to their own colonialism in the Americas. While the most outrageous parts of these claims have been debunked and shown to be larger British propaganda, they still appear regularly in pop history, notably that Oatmeal comic about Columbus that gets shared every October.

Sources: Andrew Wawn, The Vikings and the Victorians: Inventing the Old North in Nineteenth-Century Britain.

Christopher Marlow, Tamburlaine

People Magazine, The Children of John Wayne, Susan Hayward and Dick Powell Fear That Fallout Killed Their Parents.

Samuel Taylor Coleridge, Kublai Khan: or, A Vision in a Dream: A Fragment

Roy Rogers, "What the Oatmeal Missed.'