How can the Elizabethan world picture be applied to Hamlet, Macbeth, and Othello?
Well, since your question lists the general genre of tragedies as a tag, I would say that it is safe to include at least King Lear and Julius Caesar in your discussion. All the plays that you have mentioned, plus the two that I have listed, share one element: They depict war as a necessary part of settling conflict between nations or within a nation.
Shakespeare lived and wrote (for the most part) during the reign of Elizabeth I, who is considered to have been the most successful British monarch for, among other things, the military strength and sovereignty she built for her country and people. But this is not to imply that the times were peaceful. England was in an almost constant battle (fought mostly at sea) against Spain. And so, in the plays you mention, war and battling, and the strength and honor necessary to a good soldier, are prominent themes.
Elizabethans believed that being strong militarily was very important and "might", to them, often made "right." In Macbeth, there is clearly honor and respect to be earned for keeping sovereignty intact through battle.
In Othello, all the main male characters are soldiers, and their relationships are governed (and their resentments fueled) by this pecking order.
And, in Hamlet, though it is often seen as a subplot, Denmark is in the middle of a border war with Norway -- a war in which it is noted that Hamlet Senior was a brave and valiant participant. It is not a small matter in the play that Hamlet himself doesn't seem very concerned with politics or power as contrasted with Laertes, who returns to Denmark upon hearing of his father's death ready to revolt.
So, one way in which the Elizabethan world view can be applied to each of these plays is that they present an understanding of the world that insists that war is necessary to solve questions of power and conflict between nations, and that an important part of one's manhood is governed by the rules of soldierly comradeship.
The world view during Shakespeare's time can be found in many of his plays. We can, as others have mentioned, note the presence of the supernatural. This was not something that Shakespeare invented: he included it because a belief in other-worldly creatures was popular at the time: people believed whole-heartedly in the supernatural.
There was the concentration on the Great Chain of Being: that people were ordained to be in certain spots within the hierarchy of the universe. The poor were doomed to be poor. A king was ordained by God to be a king. With this in mind, if someone stepped out of his natural place and took another position—specifically killing a king and taking his place—the world view was that the universe would be out of kilter until the imbalance was somehow corrected. This would lead directly into the debate over whether anyone was ever justified to kill a king. Is Macduff justified in killing Macbeth? or Hamlet killing Claudius? or Brutus killing Caesar? It would seem not in that Hamlet and Brutus both die. However, Macduff is not killed—and this play was written by Shakespeare to honor James I of England (formerly James VI of Scotland). In writing this Scottish play for his new king, Shakespeare allows Macduff to kill Macduff, who had killed James I's ancestor, Banquo. However, while the play is factual to an extent, Macbeth (in truth) came to the throne honorably in battle. So it would seem that in some circumstances, regicide was acceptable, though this was a highly debated topic at that time.
Hamlet wrote for his audiences. His success was based upon his acceptance by his audiences. He was an author who knew how to please the paying customer, and so his work reflected the world view that people of that time accepted. To continue to write and perform his plays, Shakespeare knew he needed to draw the audience to care about his characters, and to do so, he needed not to offend Elizabethan England, but to provide them with plots that reflected things his patrons believed.
Another aspect of the Elizabethan world view would be their perception of the Kingship = the Country. The health of the king was equal to the health of the kingdom. The goodness of the king was equal to the goodness of the kingdom, etc. In literary terms, this would be called "The Chain of Being Metaphor" -- everything related to the king is chained to the being of the state. Because the king is the embodiement of God on earth, there is also a connection between kings and faith.
One of the greatest points of stress for Hamlet is his thought that Denmark is "an unweeded garden" due to the behavior of Claudius and Gertrude. This is even more so after he learns that Claudius killed for the throne. Hamlet needs to avenge the death to make right the throne of Denmark because that will make right the kingdom of Denmark.
The actions and the aftermath of Macbeth and Othello can also be interpreted in light of this metaphor. How do they as men reflect the state of their kingdoms?
I'll be a bit less specific, but two ideas come to mind. One is the use of supernatural elements, such as witches and ghosts and other what we might call superstitious beliefs. The Elizabethans were a very superstitious people and they would not have been surprised by these elements appearing on stage; in fact, they understood their import on characters and plot in a way we do not (at least not without study). The second is the disruption in nature when a God-appointed king has died of unnatural causes. Eternal damnation was the punishment for the murderer, and nature rebelled when the order of the universe, so to speak, had been disrupted.
I think the major link you can make between Hamlet and Macbeth is through a focus on superstition and on the importance of the King and Divine Rule. Macbeth in many ways has lots of links with Julius Caesar in a sustained focus on supernatural events that reflect how the country itself is being ripped apart by the thoughts of regicide or the desire to kill the leader. Likewise, the appearance of the Ghost in Hamlet does rather strongly hint at the fact that there is something "rotten" in the state of Denmark.
I agree with number 8. Shakespeare, like all authors, was a product of his environment. I don't think he shaped his world, even though he was influential, but his world did shape him. We can learn a lot about how Elizabethans looked at the world from Shakespeare’s plays.
Hierarchy is an important concept in the Elizabethan world. With the sovereign at the top, his death or ineffectiveness upsets the entire order of life. This Chain of Being was an important part of life for the Elizabethans.
I am not sure of your question, but it is the other way around; How doe the plays represent the Elizabethan world? It was the era in which Shakespeare wrote it and he could only use what he knows; therefore, the plays represent the world in which he lived.
But what about the Chain of Being applied to Othello? I think this is difficult....