To what degree is modern American culture an outgrowth of the philosophies of the Founding Fathers? How can I write an essay synthesizing some of...

I can't write your essay for you; however, I can get you started with an idea or two.  

The first thing that you have to decide is whether or not you think modern culture is or is not "an outgrowth of the philosophies of the Founding Fathers." I teach a modern media popular culture studies class, and I link a lot of modern culture's myths, values, and beliefs to early American history.  I very much believe that modern culture is an outgrowth of early America and the Founding Fathers.  

If that is your belief as well, then the next step is to pick beliefs and values that you feel were present in the late 1700s and are still present today.  The value that I chose is called "redemptive violence."  Redemptive violence is violence that is seen as “okay” or “right” to use because it is defending a core belief or value.  In other words, the world is redeemed and made better through violence.  Might makes right.  In a nutshell, the myth works like this.  Diplomacy and talking have failed to rid the city/country/world of evil; therefore, a violent and armed hero is needed to do violence upon the "bad guy" in order to restore order. 

The next step is to provide evidence, from the sources listed, that the Founding Fathers adhered to a belief in redemptive violence.  The three sources that I would use are The Declaration of Independence, "Paul Revere's Ride," and "Speech to the Virginia Convention." 

In each of those sources, there are lines that indicate that the colonists have tried to go through legal, diplomatic channels in order to address problems.  Those have failed, and the document then suggests a call to arms.  

For example, there is this section from Henry's speech.  

We have petitioned; we have remonstrated; we have supplicated; we have prostrated ourselves before the tyrannical hands of the ministry and parliament. Our petitions have been slighted; our remonstrances have produced additional violence and insult; our supplications have been disregarded; and we have been spurned, with contempt, from the foot of the throne. In vain, after these things, may we indulge the fond hope of peace and reconciliation. There is no longer any room for hope. If we wish to be free—if we mean to preserve inviolate those inestimable privileges for which we have been so long contending—if we mean not basely to abandon the noble struggle in which we have been so long engaged, and which we have pledged ourselves never to abandon until the glorious object of our contest shall be obtained, we must fight! I repeat it, sir, we must fight! An appeal to arms and to the God of Hosts is all that is left us!

Henry clearly states all of the legal avenues that have been pursued.  He says that they have all failed, and the only course of action is violence. 

Jefferson writes the same thing in his document. 

In every stage of these Oppressions We have Petitioned for Redress in the most humble terms: Our repeated Petitions have been answered only by repeated injury.... We must, therefore, acquiesce in the necessity, which denounces our Separation, and hold them, as we hold the rest of mankind, Enemies in War, in Peace Friends.

"The Midnight Ride of Paul Revere" is a more poetic call to arms than the other two, but it also shows how Revere called the colonists to arms in order to fight against the British.  It's not a poem about diplomacy.  It's a poem about coming violence and how that is necessary.  

As for how redemptive violence works in modern culture, you only have to look to the plethora of comic book movies that are popular right now.  In just about every single case, the hero or heroine character is needed because standard law enforcement and political figures are unable to combat the new threat.  The hero character then comes in and defeats the "bad guy" through violence.  Batman and the Avengers are not called in to talk the bad guy down.  Peace is restored through copious amounts of violence.  If you need a quote from a movie, I would use the line that Frank Castle speaks from the 2004 film The Punisher. 

In certain extreme situations, the law is inadequate. In order to shame its inadequacy, it is necessary to act outside the law. To pursue… natural justice. This is not vengeance. Revenge is not a valid motive, it’s an emotional response. No. Not vengeance. Punishment.