How did president theodore roosevelt's "square deal" epitomize the notion that the federal government should serve as a steward protecting the public's interests? in addition, how did the goals and reform agenda of the progressive era manifest themselves during the presidential administrations of roosevelt, taft, and wilson?

Let me handle your first question - - always good to do one question at a time here. : - )

Prior to President Theodore Roosevelt, those who preceded him in federal government had tended to side with industry leaders, expecting laborers to fall in line and do the work for the good of the companies. In 1902, when there was a particularly tense strike by coal workers, Roosevelt invited both sides (labor leaders and management leaders) to the White House to negotiate. This was an example of the way he saw the role of government leadership as "steward" to the nation, mediating on behalf of everyone's interests, not just the interests of a powerful small group. His "Square Deal" policies were aimed at making things fair and square for the general public. An example of this would be how much land was set aside under his administration as national forests, national parks, national monuments, etc. He was seeking to protect the use of the land for all Americans' interests, rather than letting corporations tear into any land or forest they wanted in order to grab natural resources.