What are the real reasons why the United States enters into military engagements? What is the role of the media in such decisions?

The problem with this question is that the United States has entered into many different military conflicts over the past two and a half centuries, and each conflict is different. Moreover, the Unites States is not some single-minded entity but a nation of over 300 million people. Each of this individuals have their own reasons for supporting or opposing the US intervention in various military conflicts. Even more importantly, people rarely have singular motivations, but often multiple different reasons for a position.

For example, someone might want to intervene in Libya due to the ongoing humanitarian crises in that country, also want to intervene because the waves of migration from people fleeing the civil war have destabilizing effects on the region, but be reluctant to intervene due to unwillingness to risk American lives, fear of escalating conflicts with Russia, and concern over the lack of a clear replacement civil authority to support. 

In World War II, to take another important example, opinions in the United States were divided until Japan's attack on Pearl Harbor in 1941. While some people wanted to intervene to stop Hitler, due to concern about his attacks on our allies and his program of ethnic cleansing, there was a strongly isolationist element in the country, and the US did not consistently welcome Jewish immigrants until 1944. The US government was actively involved in a propaganda effort to increase support for the war with the full complicity of the media. President Roosevelt's fireside chats were an example of the use of radio to encourage people to support the war effort and to keep citizenry well-informed about the progress of the war.

The US entry into the Vietnam War was more gradual and accidental, beginning as a matter of supplying aid to the French and then gradually evolving into a proxy war as part of the US opposition to the Soviet Union. Part of the reasoning was the domino theory, in which it was assumed that if one country became aligned to the Soviet bloc, others would follow. Media coverage of atrocities in the war, including the use of napalm on civilians, was instrumental in ending the war. 

In general, the reasons for military intervention vary greatly, ranging from economic to ethical and from mutual treaties and obligations to global balance of power. The media can serve to inflame popular sentiment either for or against wars.