How can I analyze Hearn's description of the cotton press at the beginning of "Inventing New Orleans"? What is its symbolic significance? What are Hearn's first impression of New Orleans as he travels on the Mississippi from Memphis?

a) Hearn's description of the cotton press and its symbolic significance.
Hearn characterizes the cotton press as a large and imposing leviathan, a machine (along with the cotton gin) that forms the backbone of the cotton industry in 19th century America. In fact, both the cotton gin and press were largely responsible for ushering in the industrial revolution to the United States. These machines revolutionized cotton production and raised the yearly American value of the crop from a typical $150,000 to more than $8 million. To ensure that production occurred at an optimal pace, slaves were utilized. Not only were they used to plant the cotton crops, they also had to drag or roll huge bales of heavy cotton to the presses.
So, the significance of the cotton press is central to Hearn's argument. It symbolizes not only progress and industry but also the entrenchment of slavery in the deep American South. In the excerpt, Hearn describes the cotton press as a frightening monster:

The spectacle of this colossal press in motion is really terrific. It is like a nightmare of iron and brass. It does not press downward but upward. It is not a press as we understand the term generally, but an enormous mouth of metal which seizes the bale and crushes it in its teeth...a monstrous head of living iron and brass... having pointed gaps in its face like Gothic eyes...a mouth five feet wide...the lower jaw alone moves...worked by two vast iron tendons, long and thick and solid as church pillars.

Hearn states that two black men (presumably slaves) had to roll a plantation bale of upwards to a thousand pounds into the mouth of the cotton press. He compares the press to one of those machines with "horrible, yawning heads...through which awful jaws the sacrificial victims passed." Essentially, Hearn characterizes the cotton press as a symbol of an industry that devours unsuspecting slaves in its grasp.
b) Brief overview of Hearn's first impression of New Orleans as he travels on the Mississippi from Memphis.
As Louisiana's largest city and one of America's major ports, New Orleans was significant even in Hearn's time. The city is located along the Mississippi River, which allows Hearn to relate his first impressions of it as he travels on the river from Memphis.
Since you require Hearn's first impression of New Orleans, I will first concentrate on his initial view of the city and then provide a brief overview of his impression.
As he approaches the New Orleans port, he notices that there are eighteen miles of levee, and this fascinates him. In the center of the wharves and piers, he describes the enormous Cotton and Sugar Landings, the imposing sugar shacks, and the rising smoke from the cotton-press furnaces. Most notably, there are all manner of sea vessels, representing the countries of the globe, gliding along the wide expanse of the river. Hearn describes:

...barks and brigs, schooners and brigantines, frigates and merchantmen, of all tonnages--ships of light and graceful build ...deep-bellied steamers with East Indian names...strong-bodied vessels from Norway and all the Scandinavian ports; tight looking packets from English ports; traders under German, Dutch, Italian, French, and Spanish flags; barks from the Mediterranean; shapely craft from West-Indian harbors...

Hearn's first impression of New Orleans is that it is a fascinating place, mainly because it reminds him of some aspect of his homeland. Hearn asserts that New Orleans "resembles no other city upon the face of the earth, yet it recalls vague memories of a hundred cities." He maintains that this is the main charm inherent in New Orleans: that it reminds each foreigner of his own homeland and of the things he treasures.