At the very beginning of most economics courses it is stated that there are four players in the economic system (you might just say in the market), there are consumers trying to maximize value, firms trying to maximize profits, the government trying to maximize welfare (obviously this is about democratically elected governments), and foreigners doing all of the above. The point here is that the government's objective is to maximize welfare. More specifically, they do this by deregulating monopolies, forbidding trusts/cartels and more that I will go on with in your next question.

2. There are basically two core parts to this, monetary policy and fiscal policy. Monetary policy is controlling the money supply, this is done in three ways, the discount rates, required reserves, and open market operations. The discount rate is the rate at which the Federal Reserve lends money to banks, if they lend it at low rates then banks will also lend at low rates (because they don't have to pay much for the money) and more people will have cash on hand (so an increase in money supply). The required reserves is the amount that a bank must keep on reserve in proportion to how much they can lend (it's much more complex but that's the idea), it basically tells banks that they can't lend too much unless they have it backed up, thereby decreasing the money supply. Open market operations is basically buying and selling bonds and securities, if the Fed buys then people have cash and the Fed has the bonds, if the Fed sells then they take cash and give you bonds. That is a very summarized version of monetary policy. The second core aspect is fiscal policy. Fiscal policy is the regulations for how much tax is collected and how much the government spends (and what they spend it on). This includes deciding if it is better to let people keep more money and spend it by themselves vs collecting more and building bridges, schools, weapons, emergency services (although emergency services is usually payed by local taxes) etc. Fiscal policy is far, far more complex than these five lines but that is the main concept.

3. The main controllers economic policy in the US are the Chairman of the Federal Reserve (in reality this is whatever central bank a country has, for us it is the Federal Reserve and its highest ranking person is the Chairman), the President, Congress, and the Secretary of the Treasury. US money basically works like this: the Secretary of the Treasury advises the President on all economic stuff and the President proposes a budget to Congress. Congress passes the budget into law (as long as they agree with it). The secretary is also the person who handles all of the US government's transactions and he needs to sign Federal Reserve notes for them to be legal tender. Something that many people are not aware of is the fact the Federal Reserve is not a government entity, it is its own entity and the only thing that it legally must do is present its policy plans to Congress. Congress can strongly disagree but there is essentially nothing that the government can do to control the Federal Reserve. In other words monetary policy is not controlled by the US government. The main reason for this is to keep it free from corruption. In a nutshell, that's the way it works.