1) One of the main features of American history in the age of industrialization was the growing presence of the federal government in American society. In an essay specifically explain why the government began to aid the expansion of the market economy during the Civil War and how the Homestead and Pacific Railroad Acts passed by the federal government during this period helped lead to the defeat of the numerous Native American nations in the West. In addition, what domestic factors (events and issues within the United States) help explain why the United States government began to build a commercial overseas empire in the 1890s? Finally, how did President Teddy Roosevelt change the relationship between the government and big business during the progressive period? Be sure to cite specific events and ideas in your answer.

It looks like there are three parts to your essay question. I will provide guidance on them below. 
1) Why the government began to aid the expansion of the market economy during the Civil War.
The main reason for the federal government's actions was to strengthen the North's newly industrialized economy. During the Civil War, the Southern states depended upon the cotton industry, one wholly supported by slave labor. The South lagged far behind the North in terms of manufacturing progress.
Due to the increased mechanization in factories in the Northern United States, states north of the Mason-Dixie line produced more firearms, railroad equipment, farm machinery, and textiles than those in the South. Most importantly, the North produced more and better weapons than the South, and this factor alone provided the North with a distinct advantage in its war against the South. The North also built its railroads faster than the South. Why? The North had access to a large labor population, from people moving into cities and foreign workers immigrating to the American North.
During the Civil War, Southern politicians elected to leave Congress to join the Confederacy. This led the federal government to push for westward expansion and to pass legislation to maintain the North's economic supremacy. Essentially, the federal government supported the emergence of a market economy so that it could strengthen its position against the South during the Civil War. In the next part, I'll address the Homestead and Pacific Railroad Acts.
Sources:
Industry and Economy During the Civil War.
The Economics of the Civil War.
How the Homestead and Pacific Railroad Acts passed by the federal government during this period helped lead to the defeat of the numerous Native American nations in the West.
The federal government planned to increase the North's economic power by exporting industrialization to the west. It projected that westward expansion was best supported by legislation such as the Homestead Act, the Morrill Act, and the Pacific Railroad Acts of 1862. For more details about these laws in your essay, please read about them at the link provided.
The transcontinental railroad basically linked the eastern and western United States, strengthening the North's position in its economic war against the South. The Homestead Act was especially significant because it allowed women and African Americans to claim land. The new pieces of legislation, however, led to the defeat of Native American tribes. Many tribes were displaced when Americans moved in to claim land in the Mid-West. The transcontinental railroad brought increased economic expansion to the west, but it also proved deleterious to many Indian tribes.
When Americans discovered that leather could be manufactured out of bison hide, the slaughter of herds of bison became commonplace. Now, Native American tribes were deprived of one of their main sources of food. Additionally, Indian warriors found themselves increasingly engaged in constant battle with the United States army; they always fared badly in these skirmishes. Read more here: American Indians and the Transcontinental Railroad.
Domestic factors (events and issues within the United States) that help explain why the United States government began to build a commercial overseas empire in the 1890s.
Two major factors may help explain why the United States government began to build a commercial overseas empire during the 1890s; these are the Panic of 1893 and the concept of Manifest Destiny.
The Panic of 1893 was the direct result of American manufacturing growth. Industrialization led to great economic growth within the United States, but the market was soon saturated. The United States made too many products, more than enough for its population.
Essentially, production exceeded consumption. The United States needed new markets to sell a glut of manufactured products. Outward expansion to foreign markets was thought to be the answer to this crisis.
The Panic of 1893 was also caused by the overexpansion of the railroad industry. Loans were made to firms that wanted to build more railroads, but many of these firms overextended themselves with huge loans. These firms eventually went bankrupt and laid off workers. People rushed to withdraw money from their banks, precipitating a panic.
The federal government stepped in to reassure the public, promising them access to more jobs. In order to keep its promise, foreign economic expansion was necessary. The recession was a major reason the United States began to build a commercial overseas empire in the 1890s.
3) For the third part of your question, I suggest referring to the link I provide below. Roosevelt basically changed the relationship between government and big business during the Progressive period. He sided with the workers against Big Business. When Congress refused to act, Roosevelt signed executive orders to tamp down on monopolistic practices by companies. He fervently believed that the federal government had the right to monitor Big Business.
The Great Regulator
http://ushistoryscene.com/article/1862-homestead-act/