In general, his reputation as a Trust Buster does hold true. In the time between the Sherman Anti-Trust Act's signing into law in 1890 and Theodore Roosevelt becoming president in 1901, the act had only been used sparingly to break up big business.

Before Theodore Roosevelt, it was not heavily used by other presidents. Using this book as a source, it would appear that under Benjamin Harrison, four or five different lawsuits were initiated under the Antitrust act.

Stephen Grover Cleveland (on his second time being president) filed 13. However, many of those were one related to one lawsuit, and two were not against a monopoly, but a Union (the Sherman Anti-trust act was against anti-competetive practices, of which Unions were considered one initially). In fact, at least a full five appear to be related to just one Union, the American Railway Union, organized in part by Eugene Victor Debs founder of the Socialist Party in America. Also noteworthy, one of the lawsuits was a near fatal blow to the law, with United States vs E C Knight Company not breaking up the company despite controlling nearly 98% of sugar production in the country. The law was still used afterwards, however.

Next under the Mckinley Administration, of whom Theodore was Vice President, only three lawsuits were filed under the act.

Then after McKinley was Assassinated, Theodore Roosevelt became president, and he filed 44 lawsuits against different trusts (although from elsewhere I have heard 54 were filed). Notably, Northern Securities Co. a railway conglomeration was broken up, and lawsuits against Standard Oil, perhaps the greatest trust of the time, were filed and eventually under the Taft administration the company was broken up (the successor companies, were Exxon and Mobil which later merged back to form what many consider the largest company today, and the now defunct Amoco Company).

I will see if I can answer some of your other questions in a second.

Edit: As for his opposition to big government and big business, it depends on what you consider to be "big government." The Bull Moose or Progressive party platform for the year 1912, for which Theodore was the nominee,
included expanding and creating new social welfare programs like federally funded healthcare and social security as well as other things like an inheritance tax, minimum wage laws and workmen's comp. However, it also included provisions to give citizens greater roles in governance such as women's suffrage, Direct election of senators (originally they were elected by state representative and senators), primary elections, recall elections, referendums (laws or government actions that people instead of law makers vote on), initiatives (petitions to bring laws to a vote), and a measure which would allow citizens to overturn a supreme court ruling.

Edit2: One important point I missed, William Howard Taft, Roosevelt's successor, actually had more lawsuits filed using the Sherman Anti-Trust Act than Theodore Roosevelt did. So although Roosevelt was the first real trust buster, Taft could be considered to be a bigger trust buster than Roosevelt. Whether Taft or Roosevelt deserves the title more is subjective really.