Was there ever a period in Russian history where their eastern frontier was kind of like America’s western frontier? Did a type of “Wild East” myth emerge from this period similar to America’s “Wild West” myth?

There was, but it wasn't the east towards Siberia, but the south into the Caucasus. After the Russian annexation of Georgia in 1801 they slowly kept increasing their presence there until 1878, when the aftermath of the most recent Russo-Turkish War saw the port city of Batumi and the interior city of Kars ceded.

Throughout these years the main occupation of the Russians was the Chechen/Dagestani Wars. While Georgia was incorporated into the Empire by 1810, and Armenia and Azerbaijan in 1828, the regions of the North Caucasus (modern Chechnya, Dagestan, Ingushetia, and parts of Cherkassia, among others), wars lasted from 1817 until 1864. In a slight parallel to the American Wild West, it was largely "uncivilized" groups that refused to recognise the authority of the larger power. In this case, it was the local ethnic groups, and unlike the American experience they were largely consolidated under one leader, Imam Shamil, an ethnic Dagestani Muslim cleric who followed Naqshbandism, a type of Sufism, which is itself a branch of Sunni Islam.

For years Russia would send soldiers down to the Caucasus to fight, with hundreds of thousands of soldiers stationed down there at its peak. Included in this was several notable writers, who developed their skills down there. Arguably the most famous would be Alexander Pushkin, who only spent a total of a few weeks in the region, but set the standard for writing about the region; in the words of Charles King "the literary pioneer to whom every later writer of poetry and prose owed some artistic debt."

After the death of Pushkin a compatriot of his was exiled to the region for writing a poem defending him. Mikhail Lermontov defined the genre of writing about the Caucasus, with his most influential work being the novel A Hero of Our Time (in Russian, Герой нашего времени). It is focused on a soldier in the region, and his various exploits. Like Pushkin, Lermontsov also died young, but he helped reform the idea of using the Caucasus as a setting for literacy.

It was also not just the Russian writers that had an influence. The themes of the region, being warfare, lawless bandits, adventures, and so on, was also a popular theme in Georgian literature. I can't think of many titles at the moment, except I will note that they were a popular topic by a young Iosef Jughashvili, who especially enjoyed reading The Patricide by Alexander Kazbegi. Jughashvili would even adopt the name of one of characters in the book, Koba, as his own name during his revolutionary career, before changing to Stalin. It's thought that Stalin identified with the bandit-like, yet noble, behaviour of Koba, and wanted to emulate him.